https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Crowley&feedformat=atomUW-Math Wiki - User contributions [en]2022-12-06T20:54:46ZUser contributionsMediaWiki 1.35.6https://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar_Spring_2022&diff=22923Graduate Algebraic Geometry Seminar Spring 20222022-03-03T20:13:12Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' 4:30-5:30 PM Thursdays<br />
<br />
'''Where:''' VV B231<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the [https://hilbert.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&redirect=yes Algebraic Geometry Seminar]. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is [https://groups.google.com/u/2/a/g-groups.wisc.edu/g/gags here].<br />
<br />
''' Organizers: ''' [https://johndcobb.github.io John Cobb]<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested, please fill out [https://forms.gle/iwvCQPKp3mDD3HZd9 this form]. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the [https://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_Algebraic_Geometry_Seminar main page].<br />
<br />
=== Spring 2022 Topic Wish List ===<br />
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.<br />
* Hilbert Schemes<br />
* Reductive groups and flag varieties<br />
* Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"<br />
* Going from line bundles and divisors to vector bundles and chern classes <br />
* A History of the Weil Conjectures<br />
* Mumford & Bayer, "What can be computed in Algebraic Geometry?" <br />
* A pre talk for any other upcoming talk<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker<br />
* Ask Questions Appropriately<br />
<br />
== Talks ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title'''<br />
|-<br />
| bgcolor="#E0E0E0"| February 10<br />
| bgcolor="#C6D46E"| Everyone<br />
| bgcolor="#BCE2FE"|[[#February 10| Informal chat session ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 17<br />
| bgcolor="#C6D46E"| Asvin G<br />
| bgcolor="#BCE2FE"|[[#February 17| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 24<br />
| bgcolor="#C6D46E"| Yu Luo<br />
| bgcolor="#BCE2FE"|[[#February 24| Riemann-Hilbert Correspondence ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 3<br />
| bgcolor="#C6D46E"| <br />
| bgcolor="#BCE2FE"|[[#March 3| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 10<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[#March 10| An introduction to Tropicalization ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 24<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[#March 24| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 31<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[#March 31| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 7<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[#April 7| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 14<br />
| bgcolor="#C6D46E"|<br />
| bgcolor="#BCE2FE"|[[#April 14| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 21<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[#April 21| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 28<br />
| bgcolor="#C6D46E"| Karan<br />
| bgcolor="#BCE2FE"|[[#April 28| Using varieties to study polynomial neural networks ]]<br />
|-<br />
| bgcolor="#E0E0E0"| May 5<br />
| bgcolor="#C6D46E"| Ellie Thieu<br />
| bgcolor="#BCE2FE"|[[#May 5| ]]<br />
|}<br />
</center><br />
<br />
=== February 10 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Everyone '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Informal chat session<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Bring your questions!<br />
|} <br />
</center><br />
<br />
=== February 17 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin G '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD<br />
|} <br />
</center><br />
<br />
=== February 24 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Yu LUO (Joey) '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Riemann-Hilbert Correspondence<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: During the talk, I will start with "nonsingular" version of Riemann-Hilbert correspondence between flat vector bundles and local systems. Then I will introduce the regular singularity, then sketch the Riemann-Hilbert correspondence with regular singularity. If time permit, I will brief mention some applications.<br />
\end{abstract}<br />
|} <br />
</center><br />
<br />
=== March 3 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:<br />
|} <br />
</center><br />
<br />
=== March 10 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Tropicalization<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Tropicalization is a logarithmic process (functor) that takes embedded algebraic varieties to polyhedral complexes. The complexes that are in the image have some additional structure which leads to the definition of a tropical variety, the main object of study in tropical geometry. I'll talk about the first paper to use these ideas, and the problem that they were used to solve.<br />
|} <br />
</center><br />
<br />
=== March 17 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== March 24 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== March 31 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 7 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 14 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 21 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 28 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Karan '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Using varieties to study polynomial neural networks<br />
<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: In this talk, I will exposit the work of Kileel, Trager, and Bruna in their 2019 paper "On the Expressive power of Polynomial Neural Networks". We will look at 1) what a polynomial neural network is and how we can interpret the output such networks as varieties, 2) why the dimension of this variety and the expressive power of this network are related, and 3) how the study of these varieties might tell us something about the architecture of the network. <br />
|} <br />
</center><br />
<br />
=== May 5 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== Past Semesters ==<br />
[https://hilbert.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2021 Fall 2021]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2021 Spring 2021]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2020 Fall 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2020 Spring 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar_Spring_2022&diff=22922Graduate Algebraic Geometry Seminar Spring 20222022-03-03T20:12:36Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' 4:30-5:30 PM Thursdays<br />
<br />
'''Where:''' VV B231<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the [https://hilbert.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&redirect=yes Algebraic Geometry Seminar]. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is [https://groups.google.com/u/2/a/g-groups.wisc.edu/g/gags here].<br />
<br />
''' Organizers: ''' [https://johndcobb.github.io John Cobb]<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested, please fill out [https://forms.gle/iwvCQPKp3mDD3HZd9 this form]. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the [https://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_Algebraic_Geometry_Seminar main page].<br />
<br />
=== Spring 2022 Topic Wish List ===<br />
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.<br />
* Hilbert Schemes<br />
* Reductive groups and flag varieties<br />
* Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"<br />
* Going from line bundles and divisors to vector bundles and chern classes <br />
* A History of the Weil Conjectures<br />
* Mumford & Bayer, "What can be computed in Algebraic Geometry?" <br />
* A pre talk for any other upcoming talk<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker<br />
* Ask Questions Appropriately<br />
<br />
== Talks ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title'''<br />
|-<br />
| bgcolor="#E0E0E0"| February 10<br />
| bgcolor="#C6D46E"| Everyone<br />
| bgcolor="#BCE2FE"|[[#February 10| Informal chat session ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 17<br />
| bgcolor="#C6D46E"| Asvin G<br />
| bgcolor="#BCE2FE"|[[#February 17| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 24<br />
| bgcolor="#C6D46E"| Yu Luo<br />
| bgcolor="#BCE2FE"|[[#February 24| Riemann-Hilbert Correspondence ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 3<br />
| bgcolor="#C6D46E"| <br />
| bgcolor="#BCE2FE"|[[#March 3| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 10<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[#March 10| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 24<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[#March 24| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 31<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[#March 31| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 7<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[#April 7| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 14<br />
| bgcolor="#C6D46E"|<br />
| bgcolor="#BCE2FE"|[[#April 14| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 21<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[#April 21| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 28<br />
| bgcolor="#C6D46E"| Karan<br />
| bgcolor="#BCE2FE"|[[#April 28| Using varieties to study polynomial neural networks ]]<br />
|-<br />
| bgcolor="#E0E0E0"| May 5<br />
| bgcolor="#C6D46E"| Ellie Thieu<br />
| bgcolor="#BCE2FE"|[[#May 5| ]]<br />
|}<br />
</center><br />
<br />
=== February 10 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Everyone '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Informal chat session<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Bring your questions!<br />
|} <br />
</center><br />
<br />
=== February 17 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin G '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD<br />
|} <br />
</center><br />
<br />
=== February 24 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Yu LUO (Joey) '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Riemann-Hilbert Correspondence<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: During the talk, I will start with "nonsingular" version of Riemann-Hilbert correspondence between flat vector bundles and local systems. Then I will introduce the regular singularity, then sketch the Riemann-Hilbert correspondence with regular singularity. If time permit, I will brief mention some applications.<br />
\end{abstract}<br />
|} <br />
</center><br />
<br />
=== March 3 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:<br />
|} <br />
</center><br />
<br />
=== March 10 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Tropicalization<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Tropicalization is a logarithmic process (functor) that takes embedded algebraic varieties to polyhedral complexes. The complexes that are in the image have some additional structure which leads to the definition of a tropical variety, the main object of study in tropical geometry. I'll talk about the first paper to use these ideas, and the problem that they were used to solve.<br />
|} <br />
</center><br />
<br />
=== March 17 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== March 24 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== March 31 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 7 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 14 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 21 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 28 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Karan '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Using varieties to study polynomial neural networks<br />
<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: In this talk, I will exposit the work of Kileel, Trager, and Bruna in their 2019 paper "On the Expressive power of Polynomial Neural Networks". We will look at 1) what a polynomial neural network is and how we can interpret the output such networks as varieties, 2) why the dimension of this variety and the expressive power of this network are related, and 3) how the study of these varieties might tell us something about the architecture of the network. <br />
|} <br />
</center><br />
<br />
=== May 5 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== Past Semesters ==<br />
[https://hilbert.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2021 Fall 2021]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2021 Spring 2021]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2020 Fall 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2020 Spring 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar_Spring_2022&diff=22769Graduate Algebraic Geometry Seminar Spring 20222022-02-16T00:28:30Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' 4:30-5:30 PM Thursdays<br />
<br />
'''Where:''' VV B231<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the [https://hilbert.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&redirect=yes Algebraic Geometry Seminar]. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is [https://groups.google.com/u/2/a/g-groups.wisc.edu/g/gags here].<br />
<br />
''' Organizers: ''' [https://johndcobb.github.io John Cobb]<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested, please fill out [https://forms.gle/iwvCQPKp3mDD3HZd9 this form]. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the [https://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_Algebraic_Geometry_Seminar main page].<br />
<br />
=== Spring 2022 Topic Wish List ===<br />
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.<br />
* Hilbert Schemes<br />
* Reductive groups and flag varieties<br />
* Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"<br />
* Going from line bundles and divisors to vector bundles and chern classes <br />
* A History of the Weil Conjectures<br />
* Mumford & Bayer, "What can be computed in Algebraic Geometry?" <br />
* A pre talk for any other upcoming talk<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker<br />
* Ask Questions Appropriately<br />
<br />
== Talks ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title'''<br />
|-<br />
| bgcolor="#E0E0E0"| February 10<br />
| bgcolor="#C6D46E"| Everyone<br />
| bgcolor="#BCE2FE"|[[#February 10| Informal chat session ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 17<br />
| bgcolor="#C6D46E"| Asvin G<br />
| bgcolor="#BCE2FE"|[[#February 17| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 24<br />
| bgcolor="#C6D46E"| Yu Luo<br />
| bgcolor="#BCE2FE"|[[#February 24| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 3<br />
| bgcolor="#C6D46E"| <br />
| bgcolor="#BCE2FE"|[[#March 3| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 10<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[#March 10| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 24<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[#March 24| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 31<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[#March 31| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 7<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[#April 7| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 14<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[#April 14| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 21<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[#April 21| ]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 28<br />
| bgcolor="#C6D46E"| Karan<br />
| bgcolor="#BCE2FE"|[[#April 28| Using varieties to study polynomial neural networks ]]<br />
|-<br />
| bgcolor="#E0E0E0"| May 5<br />
| bgcolor="#C6D46E"| <br />
| bgcolor="#BCE2FE"|[[#May 5| ]]<br />
|}<br />
</center><br />
<br />
=== February 10 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Everyone '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Informal chat session<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Bring your questions!<br />
|} <br />
</center><br />
<br />
=== February 17 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin G '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD<br />
|} <br />
</center><br />
<br />
=== February 24 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:<br />
|} <br />
</center><br />
<br />
=== March 3 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:<br />
|} <br />
</center><br />
<br />
=== March 10 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== March 17 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== March 24 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== March 31 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 7 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 14 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 21 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
=== April 28 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Karan '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Using varieties to study polynomial neural networks<br />
<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: In this talk, I will exposit the work of Kileel, Trager, and Bruna in their 2019 paper "On the Expressive power of Polynomial Neural Networks". We will look at 1) what a polynomial neural network is and how we can interpret the output such networks as varieties, 2) why the dimension of this variety and the expressive power of this network are related, and 3) how the study of these varieties might tell us something about the architecture of the network. <br />
|} <br />
</center><br />
<br />
=== May 5 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== Past Semesters ==<br />
[https://hilbert.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2021 Fall 2021]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2021 Spring 2021]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2020 Fall 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2020 Spring 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar_Fall_2021&diff=22109Graduate Algebraic Geometry Seminar Fall 20212021-11-08T15:58:37Z<p>Crowley: /* November 11 */</p>
<hr />
<div>'''<br />
'''When:''' 5:00-6:00 PM Thursdays<br />
<br />
'''Where:''' TBD<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the [https://hilbert.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&redirect=yes Algebraic Geometry Seminar]. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is [https://groups.google.com/u/2/a/g-groups.wisc.edu/g/gags here].<br />
<br />
''' Organizers: ''' [https://johndcobb.github.io John Cobb], [https://sites.google.com/view/colincrowley/home Colin Crowley].<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested, please fill out [https://forms.gle/sa3ARndYSkBhT6LR9 this form]. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the [https://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_Algebraic_Geometry_Seminar main page].<br />
<br />
=== Fall 2021 Topic Wish List ===<br />
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.<br />
* Stacks for Kindergarteners<br />
* Motives for Kindergarteners<br />
* Applications of Beilinson resolution of the diagonal, Fourier Mukai transforms in general<br />
* Wth did June Huh do and what is combinatorial hodge theory?<br />
* Computing things about Toric varieties<br />
* Reductive groups and flag varieties<br />
* Introduction to arithmetic geometry -- what are some big picture ideas of what "goes wrong" when not over an algebraically closed field?<br />
* Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"<br />
* Going from line bundles and divisors to vector bundles and chern classes<br />
* A History of the Weil Conjectures<br />
* Mumford & Bayer, "What can be computed in Algebraic Geometry?" <br />
* A pre talk for any other upcoming talk<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker<br />
* Ask Questions Appropriately<br />
<br />
==Talks==<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| September 30<br />
| bgcolor="#C6D46E"| Yifan Wei<br />
| bgcolor="#BCE2FE"|[[#September 30| On Chow groups and K groups]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 7<br />
| bgcolor="#C6D46E"| Owen Goff<br />
| bgcolor="#BCE2FE"|[[#October 7 | Roguish Noncommutativity and the Onsager Algebra]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 14<br />
| bgcolor="#C6D46E"| Peter YI WEI<br />
| bgcolor="#BCE2FE"|[[#October 14 | Pathologies in Algebraic Geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 21<br />
| bgcolor="#C6D46E"| Asvin G<br />
| bgcolor="#BCE2FE"|[[#October 21 | Introduction to Arithmetic Schemes]] <br />
|-<br />
| bgcolor="#E0E0E0"| October 28<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[#October 28 | Classifying Varieties of Minimal Degree]] <br />
|-<br />
| bgcolor="#E0E0E0"| November 4<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[#November 4 | Syzygies and Koszul Cohomology]] <br />
|-<br />
| bgcolor="#E0E0E0"| November 11<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"| [[#November 11 | Introduction to Geometric Invariant Theory]] <br />
|-<br />
| bgcolor="#E0E0E0"| November 18<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[#November 18 | Combinatorial Hodge Theory]] <br />
|-<br />
| bgcolor="#E0E0E0"| December 2<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[#December 2 | Galois Descent]] <br />
|-<br />
| bgcolor="#E0E0E0"| December 9<br />
| bgcolor="#C6D46E"| Yu Luo<br />
| bgcolor="#BCE2FE"|[[#December 9 | Stacks for Kindergarteners]] <br />
|}<br />
</center><br />
<br />
=== September 30 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Yifan Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: On Chow groups and K groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
We define Chow groups and K groups for non-singular varieties, illustrate some basic properties, and explain how intersection theory is done using K groups (on a smooth surface). Then we proceed to compute the K group of a non-singular curve. On higher dimensions there might be some issues, if time permits we will show how these issues can be mitigated, and why Grothendieck-Riemann-Roch is one of the greatest theorems in algebraic geometry (in my humble opinion).<br />
<br />
|} <br />
</center><br />
<br />
=== October 7 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Owen Goff'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Roguish Noncommutativity and the Onsager Algebra<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
While throughout algebraic geometry and many other fields we like commutative rings, we often wonder what happens if our ring is not commutative. Say, for instance, you have A^2, but instead of xy=yx you have a relation xy = qyx for some constant q. In this talk I will discuss the consequences of this relation and how it relates to an object of combinatorial nature called the q-Onsager algebra.<br />
<br />
|} <br />
</center><br />
<br />
=== October 14 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter YI WEI'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Pathologies in Algebraic Geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: This talk serves as a brief discussion on pathologies in algebraic geometry, inspired by a short thread of Daniel Litt’s twitter. No hard preliminaries! :)<br />
<br />
|} <br />
</center><br />
<br />
=== October 21 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin G'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Introduction to Arithmetic Schemes<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Many of us are comfortable working with varieties over the complex numbers (or other fields) but part of the magic is that it's almost as easy to consider varieties over more exotic rings like the integers or the p-adics.<br />
<br />
I'll explain how to think about such varieties and then use them to prove the birational invariance of Hodge numbers for Calabi-Yau's over the complex numbers using results from finite fields and p-adic analysis!<br />
<br />
TBD<br />
<br />
|} <br />
</center><br />
<br />
=== October 28 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Classifying Varieties of Minimal Degree<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
The degree of a variety embedded in projective space is a well-defined invariant, and there is a sense in which some varieties have minimal degree. Long ago, Del Pezzo and Bertini classified geometrically all possible projective varieties of minimal degree. More recently, Eisenbud and Goto gave an algebraic notion that classifies such varieties. In this talk, we will introduce the necessary background and explore these two theorems and the ways they are connected.<br />
<br />
|} <br />
</center><br />
<br />
=== November 4 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Syzygies and Koszul Cohomology<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
Early on in the history of algebraic geometry it was recognized that many properties/invariants of projective varieties could be deduced by looking at their hyperplane sections. Starting in the 1950s, this classical picture was gradually refined into general theory by people like Serre and Kodaira — many hard-earned numbers could now be obtained by more brainless methods. I hope to motivate a few ideas introduced in the 1980’s as a continuation of this story beginning from Serre’s vanishing theorem.<br />
|} <br />
</center><br />
<br />
=== November 11 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Introduction to Geometric Invariant Theory<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
Given a group action on a variety, is there a quotient variety? How do you construct it? Geometric invariant theory gives partial answers to these questions for projective varieties and a particular class of groups (reductive groups). I’ll give an overview of how GIT quotients work, which will be in the language of Hartshorne chapter one and does not require any knowledge of schemes. (Although I may need to talk a little about ample line bundles. I haven't decided yet.)<br />
<br />
With the remaining time I'll sketch how these ideas are used in constructing (coarse) moduli spaces of semistable vector bundles, and mention which areas of math use these ideas today.<br />
|} <br />
</center><br />
<br />
=== November 18 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Combinatorial Hodge Theory<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: TBD<br />
<br />
|} <br />
</center><br />
<br />
=== December 2 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Mine'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Galois Descent<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
TBD<br />
<br />
|} <br />
</center><br />
<br />
=== December 9 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Yu Luo'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Stacks for Kindergarteners<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Brief introduction to stacks.<br />
|} <br />
</center><br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2021 Spring 2021]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2020 Fall 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2020 Spring 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar_Fall_2021&diff=22103Graduate Algebraic Geometry Seminar Fall 20212021-11-08T15:51:43Z<p>Crowley: /* November 11 */</p>
<hr />
<div>'''<br />
'''When:''' 5:00-6:00 PM Thursdays<br />
<br />
'''Where:''' TBD<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the [https://hilbert.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&redirect=yes Algebraic Geometry Seminar]. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is [https://groups.google.com/u/2/a/g-groups.wisc.edu/g/gags here].<br />
<br />
''' Organizers: ''' [https://johndcobb.github.io John Cobb], [https://sites.google.com/view/colincrowley/home Colin Crowley].<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested, please fill out [https://forms.gle/sa3ARndYSkBhT6LR9 this form]. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the [https://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_Algebraic_Geometry_Seminar main page].<br />
<br />
=== Fall 2021 Topic Wish List ===<br />
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.<br />
* Stacks for Kindergarteners<br />
* Motives for Kindergarteners<br />
* Applications of Beilinson resolution of the diagonal, Fourier Mukai transforms in general<br />
* Wth did June Huh do and what is combinatorial hodge theory?<br />
* Computing things about Toric varieties<br />
* Reductive groups and flag varieties<br />
* Introduction to arithmetic geometry -- what are some big picture ideas of what "goes wrong" when not over an algebraically closed field?<br />
* Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"<br />
* Going from line bundles and divisors to vector bundles and chern classes<br />
* A History of the Weil Conjectures<br />
* Mumford & Bayer, "What can be computed in Algebraic Geometry?" <br />
* A pre talk for any other upcoming talk<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker<br />
* Ask Questions Appropriately<br />
<br />
==Talks==<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| September 30<br />
| bgcolor="#C6D46E"| Yifan Wei<br />
| bgcolor="#BCE2FE"|[[#September 30| On Chow groups and K groups]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 7<br />
| bgcolor="#C6D46E"| Owen Goff<br />
| bgcolor="#BCE2FE"|[[#October 7 | Roguish Noncommutativity and the Onsager Algebra]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 14<br />
| bgcolor="#C6D46E"| Peter YI WEI<br />
| bgcolor="#BCE2FE"|[[#October 14 | Pathologies in Algebraic Geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 21<br />
| bgcolor="#C6D46E"| Asvin G<br />
| bgcolor="#BCE2FE"|[[#October 21 | Introduction to Arithmetic Schemes]] <br />
|-<br />
| bgcolor="#E0E0E0"| October 28<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[#October 28 | Classifying Varieties of Minimal Degree]] <br />
|-<br />
| bgcolor="#E0E0E0"| November 4<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[#November 4 | Syzygies and Koszul Cohomology]] <br />
|-<br />
| bgcolor="#E0E0E0"| November 11<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"| [[#November 11 | Introduction to Geometric Invariant Theory]] <br />
|-<br />
| bgcolor="#E0E0E0"| November 18<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[#November 18 | Combinatorial Hodge Theory]] <br />
|-<br />
| bgcolor="#E0E0E0"| December 2<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[#December 2 | Galois Descent]] <br />
|-<br />
| bgcolor="#E0E0E0"| December 9<br />
| bgcolor="#C6D46E"| Yu Luo<br />
| bgcolor="#BCE2FE"|[[#December 9 | Stacks for Kindergarteners]] <br />
|}<br />
</center><br />
<br />
=== September 30 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Yifan Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: On Chow groups and K groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
We define Chow groups and K groups for non-singular varieties, illustrate some basic properties, and explain how intersection theory is done using K groups (on a smooth surface). Then we proceed to compute the K group of a non-singular curve. On higher dimensions there might be some issues, if time permits we will show how these issues can be mitigated, and why Grothendieck-Riemann-Roch is one of the greatest theorems in algebraic geometry (in my humble opinion).<br />
<br />
|} <br />
</center><br />
<br />
=== October 7 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Owen Goff'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Roguish Noncommutativity and the Onsager Algebra<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
While throughout algebraic geometry and many other fields we like commutative rings, we often wonder what happens if our ring is not commutative. Say, for instance, you have A^2, but instead of xy=yx you have a relation xy = qyx for some constant q. In this talk I will discuss the consequences of this relation and how it relates to an object of combinatorial nature called the q-Onsager algebra.<br />
<br />
|} <br />
</center><br />
<br />
=== October 14 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter YI WEI'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Pathologies in Algebraic Geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: This talk serves as a brief discussion on pathologies in algebraic geometry, inspired by a short thread of Daniel Litt’s twitter. No hard preliminaries! :)<br />
<br />
|} <br />
</center><br />
<br />
=== October 21 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin G'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Introduction to Arithmetic Schemes<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Many of us are comfortable working with varieties over the complex numbers (or other fields) but part of the magic is that it's almost as easy to consider varieties over more exotic rings like the integers or the p-adics.<br />
<br />
I'll explain how to think about such varieties and then use them to prove the birational invariance of Hodge numbers for Calabi-Yau's over the complex numbers using results from finite fields and p-adic analysis!<br />
<br />
TBD<br />
<br />
|} <br />
</center><br />
<br />
=== October 28 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Classifying Varieties of Minimal Degree<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
The degree of a variety embedded in projective space is a well-defined invariant, and there is a sense in which some varieties have minimal degree. Long ago, Del Pezzo and Bertini classified geometrically all possible projective varieties of minimal degree. More recently, Eisenbud and Goto gave an algebraic notion that classifies such varieties. In this talk, we will introduce the necessary background and explore these two theorems and the ways they are connected.<br />
<br />
|} <br />
</center><br />
<br />
=== November 4 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Syzygies and Koszul Cohomology<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
Early on in the history of algebraic geometry it was recognized that many properties/invariants of projective varieties could be deduced by looking at their hyperplane sections. Starting in the 1950s, this classical picture was gradually refined into general theory by people like Serre and Kodaira — many hard-earned numbers could now be obtained by more brainless methods. I hope to motivate a few ideas introduced in the 1980’s as a continuation of this story beginning from Serre’s vanishing theorem.<br />
|} <br />
</center><br />
<br />
=== November 11 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Introduction to Geometric Invariant Theory<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
Given a group action on a variety X, is there a quotient variety? How do you construct it? Geometric invariant theory gives partial answers to these questions for projective varieties and a particular class of groups (reductive groups). I’ll give an overview of how GIT quotients work, which will be in the language of Hartshorne chapter one and does not require any knowledge of schemes. (Although I may need to talk a little about ample line bundles. I haven't decided yet.)<br />
<br />
With the remaining time I'll sketch how these ideas are used in constructing (coarse) moduli spaces of semistable vector bundles, and mention which areas of math use these ideas today.<br />
|} <br />
</center><br />
<br />
=== November 18 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Combinatorial Hodge Theory<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: TBD<br />
<br />
|} <br />
</center><br />
<br />
=== December 2 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Mine'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Galois Descent<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
TBD<br />
<br />
|} <br />
</center><br />
<br />
=== December 9 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Yu Luo'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Stacks for Kindergarteners<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Brief introduction to stacks.<br />
|} <br />
</center><br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2021 Spring 2021]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2020 Fall 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2020 Spring 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar_Fall_2021&diff=22087Graduate Algebraic Geometry Seminar Fall 20212021-11-05T19:35:36Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' 5:00-6:00 PM Thursdays<br />
<br />
'''Where:''' TBD<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the [https://hilbert.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&redirect=yes Algebraic Geometry Seminar]. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is [https://groups.google.com/u/2/a/g-groups.wisc.edu/g/gags here].<br />
<br />
''' Organizers: ''' [https://johndcobb.github.io John Cobb], [https://sites.google.com/view/colincrowley/home Colin Crowley].<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested, please fill out [https://forms.gle/sa3ARndYSkBhT6LR9 this form]. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the [https://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_Algebraic_Geometry_Seminar main page].<br />
<br />
=== Fall 2021 Topic Wish List ===<br />
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.<br />
* Stacks for Kindergarteners<br />
* Motives for Kindergarteners<br />
* Applications of Beilinson resolution of the diagonal, Fourier Mukai transforms in general<br />
* Wth did June Huh do and what is combinatorial hodge theory?<br />
* Computing things about Toric varieties<br />
* Reductive groups and flag varieties<br />
* Introduction to arithmetic geometry -- what are some big picture ideas of what "goes wrong" when not over an algebraically closed field?<br />
* Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"<br />
* Going from line bundles and divisors to vector bundles and chern classes<br />
* A History of the Weil Conjectures<br />
* Mumford & Bayer, "What can be computed in Algebraic Geometry?" <br />
* A pre talk for any other upcoming talk<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker<br />
* Ask Questions Appropriately<br />
<br />
==Talks==<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| September 30<br />
| bgcolor="#C6D46E"| Yifan Wei<br />
| bgcolor="#BCE2FE"|[[#September 30| On Chow groups and K groups]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 7<br />
| bgcolor="#C6D46E"| Owen Goff<br />
| bgcolor="#BCE2FE"|[[#October 7 | Roguish Noncommutativity and the Onsager Algebra]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 14<br />
| bgcolor="#C6D46E"| Peter YI WEI<br />
| bgcolor="#BCE2FE"|[[#October 14 | Pathologies in Algebraic Geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 21<br />
| bgcolor="#C6D46E"| Asvin G<br />
| bgcolor="#BCE2FE"|[[#October 21 | Introduction to Arithmetic Schemes]] <br />
|-<br />
| bgcolor="#E0E0E0"| October 28<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[#October 28 | Classifying Varieties of Minimal Degree]] <br />
|-<br />
| bgcolor="#E0E0E0"| November 4<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[#November 4 | Syzygies and Koszul Cohomology]] <br />
|-<br />
| bgcolor="#E0E0E0"| November 11<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"| [[#November 11 | Introduction to Geometric Invariant Theory]] <br />
|-<br />
| bgcolor="#E0E0E0"| November 18<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[#November 18 | Combinatorial Hodge Theory]] <br />
|-<br />
| bgcolor="#E0E0E0"| December 2<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[#December 2 | Galois Descent]] <br />
|-<br />
| bgcolor="#E0E0E0"| December 9<br />
| bgcolor="#C6D46E"| Yu Luo<br />
| bgcolor="#BCE2FE"|[[#December 9 | Stacks for Kindergarteners]] <br />
|}<br />
</center><br />
<br />
=== September 30 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Yifan Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: On Chow groups and K groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
We define Chow groups and K groups for non-singular varieties, illustrate some basic properties, and explain how intersection theory is done using K groups (on a smooth surface). Then we proceed to compute the K group of a non-singular curve. On higher dimensions there might be some issues, if time permits we will show how these issues can be mitigated, and why Grothendieck-Riemann-Roch is one of the greatest theorems in algebraic geometry (in my humble opinion).<br />
<br />
|} <br />
</center><br />
<br />
=== October 7 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Owen Goff'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Roguish Noncommutativity and the Onsager Algebra<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
While throughout algebraic geometry and many other fields we like commutative rings, we often wonder what happens if our ring is not commutative. Say, for instance, you have A^2, but instead of xy=yx you have a relation xy = qyx for some constant q. In this talk I will discuss the consequences of this relation and how it relates to an object of combinatorial nature called the q-Onsager algebra.<br />
<br />
|} <br />
</center><br />
<br />
=== October 14 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter YI WEI'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Pathologies in Algebraic Geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: This talk serves as a brief discussion on pathologies in algebraic geometry, inspired by a short thread of Daniel Litt’s twitter. No hard preliminaries! :)<br />
<br />
|} <br />
</center><br />
<br />
=== October 21 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin G'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Introduction to Arithmetic Schemes<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Many of us are comfortable working with varieties over the complex numbers (or other fields) but part of the magic is that it's almost as easy to consider varieties over more exotic rings like the integers or the p-adics.<br />
<br />
I'll explain how to think about such varieties and then use them to prove the birational invariance of Hodge numbers for Calabi-Yau's over the complex numbers using results from finite fields and p-adic analysis!<br />
<br />
TBD<br />
<br />
|} <br />
</center><br />
<br />
=== October 28 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Classifying Varieties of Minimal Degree<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
The degree of a variety embedded in projective space is a well-defined invariant, and there is a sense in which some varieties have minimal degree. Long ago, Del Pezzo and Bertini classified geometrically all possible projective varieties of minimal degree. More recently, Eisenbud and Goto gave an algebraic notion that classifies such varieties. In this talk, we will introduce the necessary background and explore these two theorems and the ways they are connected.<br />
<br />
|} <br />
</center><br />
<br />
=== November 4 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Syzygies and Koszul Cohomology<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
Early on in the history of algebraic geometry it was recognized that many properties/invariants of projective varieties could be deduced by looking at their hyperplane sections. Starting in the 1950s, this classical picture was gradually refined into general theory by people like Serre and Kodaira — many hard-earned numbers could now be obtained by more brainless methods. I hope to motivate a few ideas introduced in the 1980’s as a continuation of this story beginning from Serre’s vanishing theorem.<br />
|} <br />
</center><br />
<br />
=== November 11 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Introduction to Geometric Invariant Theory<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given a group action on a variety X, is there a quotient variety? How do you construct it? Geometric invariant theory gives partial answers to these questions for projective varieties and a particular class of groups (reductive groups). I’ll give an overview of how GIT quotients work (this will assume Hartshorn chapter one, and some facts about line bundles) and if time permits I’ll sketch an application: constructing moduli spaces of stable vector bundles. <br />
<br />
|} <br />
</center><br />
<br />
=== November 18 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Combinatorial Hodge Theory<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: TBD<br />
<br />
|} <br />
</center><br />
<br />
=== December 2 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Mine'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Galois Descent<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
<br />
TBD<br />
<br />
|} <br />
</center><br />
<br />
=== December 9 ===<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Yu Luo'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Stacks for Kindergarteners<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Brief introduction to stacks.<br />
|} <br />
</center><br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2021 Spring 2021]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2020 Fall 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2020 Spring 2020]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=20844Graduate Algebraic Geometry Seminar2021-02-15T17:58:45Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Thursday 5:00-6:00 PM CST<br />
<br />
'''Where:''' https://uwmadison.zoom.us/j/92877740706?pwd=OVo0QmxRVEdUQ3RnUWpoWmFRRUI3dz09<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
== Spring 2021 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#E0E0E0"| February 4<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 4| A Bertini type theorem via probability]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 25<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Alex Hoff<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Roufan Jiang<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Chiahui (Wendy) Cheng<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Erika Pirnes<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Reconstruction conjecture in graph theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| TBD]]<br />
|}<br />
</center><br />
<br />
== February 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A Bertini type theorem via probability<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will prove that most hyperplane slices are irreducible over any field by reducing to finite fields and applying probabilistic arguments. The talk will be very elementary! <br />
|} <br />
</center><br />
== February 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TDB <br />
|} <br />
</center><br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD <br />
|} <br />
</center><br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Roufan Jiang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD <br />
|} <br />
</center><br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Chiahui (Wendy) Cheng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD <br />
|} <br />
</center><br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Erika Pirnes'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Reconstruction conjecture in graph theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD<br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD<br />
|} <br />
</center><br />
<br />
<br />
== Fall 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#E0E0E0"| September 30<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 30| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 5<br />
| bgcolor="#C6D46E"| Yifan Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 5| On the Analytic Side (GAGA)]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 14<br />
| bgcolor="#C6D46E"| Owen Goff<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 14| The Magic and Comagic of Hopf Algebras]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 21<br />
| bgcolor="#C6D46E"| Roufan Jiang<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 21| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 28<br />
| bgcolor="#C6D46E"| Erika Pirnes<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 28| Introduction to representation theory via an example]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 4<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 4| The Internal Language of the Category of Sheaves]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 11<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 11| Introduction to Boij-Söderberg Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 18<br />
| bgcolor="#C6D46E"| Yunfan He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 18| Introduction to mixed Hodge structure]]<br />
|-<br />
| bgcolor="#E0E0E0"| November November 25<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 2<br />
| bgcolor="#C6D46E"| Peter Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 2| K3 Surfaces and Their Moduli]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 9<br />
| bgcolor="#C6D46E"| Wendy Cheng<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 9| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 16<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 16| Characteristic Dependence of Syzygies of Random Monomial Ideals]]<br />
|}<br />
</center><br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Braid group action on derived categories]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Birational geometry: existence of rational curves]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.<br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.<br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Stanley-Reisner Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.<br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Braid group action on derived category<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: In this talk, I will define spherical object and A_m-configuration in derived category of coherent sheaves, and say a few words about the motivation coming from the homological mirror symmetry.<br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Early on in the semester, Colin told us a bit about Morse<br />
Theory, and how it lets us get a handle on the (classical) topology of<br />
smooth complex varieties. As we all know, however, not everything in<br />
life goes smoothly, and so too in algebraic geometry. Singular<br />
varieties, when given the classical topology, are not manifolds, but<br />
they can be described in terms of manifolds by means of something called<br />
a Whitney stratification. This allows us to develop a version of Morse<br />
Theory that applies to singular spaces (and also, with a bit of work, to<br />
smooth spaces that fail to be nice in other ways, like non-compact<br />
manifolds!), called Stratified Morse Theory. After going through the<br />
appropriate definitions and briefly reviewing the results of classical<br />
Morse Theory, we'll discuss the so-called Main Theorem of Stratified<br />
Morse Theory and survey some of its consequences.<br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Birational geometry: existence of rational curves<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight. <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=20838Graduate Algebraic Geometry Seminar2021-02-15T02:24:45Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Thursday 5:00-6:00 PM EST<br />
<br />
'''Where:''' https://uwmadison.zoom.us/j/92877740706?pwd=OVo0QmxRVEdUQ3RnUWpoWmFRRUI3dz09<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
== Spring 2021 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#E0E0E0"| February 4<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 4| A Bertini type theorem via probability]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 25<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Alex Hoff<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Roufan Jiang<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Chiahui (Wendy) Cheng<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Erika Pirnes<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| TBD]]<br />
|}<br />
</center><br />
<br />
== February 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A Bertini type theorem via probability<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will prove that most hyperplane slices are irreducible over any field by reducing to finite fields and applying probabilistic arguments. The talk will be very elementary! <br />
|} <br />
</center><br />
== February 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TDB <br />
|} <br />
</center><br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD <br />
|} <br />
</center><br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Roufan Jiang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD <br />
|} <br />
</center><br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Chiahui (Wendy) Cheng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD <br />
|} <br />
</center><br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Erika Pirnes'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD<br />
|} <br />
</center><br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD<br />
|} <br />
</center><br />
<br />
<br />
== Fall 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#E0E0E0"| September 30<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 30| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 5<br />
| bgcolor="#C6D46E"| Yifan Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 5| On the Analytic Side (GAGA)]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 14<br />
| bgcolor="#C6D46E"| Owen Goff<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 14| The Magic and Comagic of Hopf Algebras]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 21<br />
| bgcolor="#C6D46E"| Roufan Jiang<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 21| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 28<br />
| bgcolor="#C6D46E"| Erika Pirnes<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 28| Introduction to representation theory via an example]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 4<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 4| The Internal Language of the Category of Sheaves]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 11<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 11| Introduction to Boij-Söderberg Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 18<br />
| bgcolor="#C6D46E"| Yunfan He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 18| Introduction to mixed Hodge structure]]<br />
|-<br />
| bgcolor="#E0E0E0"| November November 25<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 2<br />
| bgcolor="#C6D46E"| Peter Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 2| K3 Surfaces and Their Moduli]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 9<br />
| bgcolor="#C6D46E"| Wendy Cheng<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 9| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 16<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 16| Characteristic Dependence of Syzygies of Random Monomial Ideals]]<br />
|}<br />
</center><br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Braid group action on derived categories]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Birational geometry: existence of rational curves]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.<br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.<br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Stanley-Reisner Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.<br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Braid group action on derived category<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: In this talk, I will define spherical object and A_m-configuration in derived category of coherent sheaves, and say a few words about the motivation coming from the homological mirror symmetry.<br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Early on in the semester, Colin told us a bit about Morse<br />
Theory, and how it lets us get a handle on the (classical) topology of<br />
smooth complex varieties. As we all know, however, not everything in<br />
life goes smoothly, and so too in algebraic geometry. Singular<br />
varieties, when given the classical topology, are not manifolds, but<br />
they can be described in terms of manifolds by means of something called<br />
a Whitney stratification. This allows us to develop a version of Morse<br />
Theory that applies to singular spaces (and also, with a bit of work, to<br />
smooth spaces that fail to be nice in other ways, like non-compact<br />
manifolds!), called Stratified Morse Theory. After going through the<br />
appropriate definitions and briefly reviewing the results of classical<br />
Morse Theory, we'll discuss the so-called Main Theorem of Stratified<br />
Morse Theory and survey some of its consequences.<br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Birational geometry: existence of rational curves<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight. <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=20837Graduate Algebraic Geometry Seminar2021-02-15T02:23:30Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:00pm<br />
<br />
'''Where:''' https://uwmadison.zoom.us/j/92877740706?pwd=OVo0QmxRVEdUQ3RnUWpoWmFRRUI3dz09<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
== Spring 2021 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#E0E0E0"| February 4<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 4| A Bertini type theorem via probability]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 25<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Alex Hoff<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Roufan Jiang<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Chiahui (Wendy) Cheng<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Erika Pirnes<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| TBD]]<br />
|}<br />
</center><br />
<br />
== February 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A Bertini type theorem via probability<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will prove that most hyperplane slices are irreducible over any field by reducing to finite fields and applying probabilistic arguments. The talk will be very elementary! <br />
|} <br />
</center><br />
== February 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TDB <br />
|} <br />
</center><br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD <br />
|} <br />
</center><br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Roufan Jiang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD <br />
|} <br />
</center><br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Chiahui (Wendy) Cheng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD <br />
|} <br />
</center><br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Erika Pirnes'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD<br />
|} <br />
</center><br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: TBD<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: TBD<br />
|} <br />
</center><br />
<br />
<br />
== Fall 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#E0E0E0"| September 30<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 30| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 5<br />
| bgcolor="#C6D46E"| Yifan Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 5| On the Analytic Side (GAGA)]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 14<br />
| bgcolor="#C6D46E"| Owen Goff<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 14| The Magic and Comagic of Hopf Algebras]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 21<br />
| bgcolor="#C6D46E"| Roufan Jiang<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 21| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 28<br />
| bgcolor="#C6D46E"| Erika Pirnes<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 28| Introduction to representation theory via an example]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 4<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 4| The Internal Language of the Category of Sheaves]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 11<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 11| Introduction to Boij-Söderberg Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 18<br />
| bgcolor="#C6D46E"| Yunfan He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 18| Introduction to mixed Hodge structure]]<br />
|-<br />
| bgcolor="#E0E0E0"| November November 25<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 2<br />
| bgcolor="#C6D46E"| Peter Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 2| K3 Surfaces and Their Moduli]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 9<br />
| bgcolor="#C6D46E"| Wendy Cheng<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 9| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 16<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 16| Characteristic Dependence of Syzygies of Random Monomial Ideals]]<br />
|}<br />
</center><br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Braid group action on derived categories]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Birational geometry: existence of rational curves]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.<br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.<br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Stanley-Reisner Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.<br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Braid group action on derived category<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: In this talk, I will define spherical object and A_m-configuration in derived category of coherent sheaves, and say a few words about the motivation coming from the homological mirror symmetry.<br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Early on in the semester, Colin told us a bit about Morse<br />
Theory, and how it lets us get a handle on the (classical) topology of<br />
smooth complex varieties. As we all know, however, not everything in<br />
life goes smoothly, and so too in algebraic geometry. Singular<br />
varieties, when given the classical topology, are not manifolds, but<br />
they can be described in terms of manifolds by means of something called<br />
a Whitney stratification. This allows us to develop a version of Morse<br />
Theory that applies to singular spaces (and also, with a bit of work, to<br />
smooth spaces that fail to be nice in other ways, like non-compact<br />
manifolds!), called Stratified Morse Theory. After going through the<br />
appropriate definitions and briefly reviewing the results of classical<br />
Morse Theory, we'll discuss the so-called Main Theorem of Stratified<br />
Morse Theory and survey some of its consequences.<br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Birational geometry: existence of rational curves<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight. <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=20126Graduate Algebraic Geometry Seminar2020-10-13T16:30:21Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:00pm<br />
<br />
'''Where:''' https://uwmadison.zoom.us/j/92877740706?pwd=OVo0QmxRVEdUQ3RnUWpoWmFRRUI3dz09<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
== Fall 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#E0E0E0"| September 30<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 30| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 5<br />
| bgcolor="#C6D46E"| Yifan Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 5| On the Analytic Side (GAGA)]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 14<br />
| bgcolor="#C6D46E"| Owen Goff<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 14| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 21<br />
| bgcolor="#C6D46E"| Roufan Jiang<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 21| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 28<br />
| bgcolor="#C6D46E"| Erika Pirnes<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 28| Introduction to representation theory via an example]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 4<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 4| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 11<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 11| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 18<br />
| bgcolor="#C6D46E"| Yunfan He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 18| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| November November 25<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 2<br />
| bgcolor="#C6D46E"| Peter Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 2| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 9<br />
| bgcolor="#C6D46E"| Wendy Cheng<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 9| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 16<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 16| TBD]]<br />
|}<br />
</center><br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Birational geometry: existence of rational curves]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.<br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.<br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Stanley-Reisner Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.<br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Early on in the semester, Colin told us a bit about Morse<br />
Theory, and how it lets us get a handle on the (classical) topology of<br />
smooth complex varieties. As we all know, however, not everything in<br />
life goes smoothly, and so too in algebraic geometry. Singular<br />
varieties, when given the classical topology, are not manifolds, but<br />
they can be described in terms of manifolds by means of something called<br />
a Whitney stratification. This allows us to develop a version of Morse<br />
Theory that applies to singular spaces (and also, with a bit of work, to<br />
smooth spaces that fail to be nice in other ways, like non-compact<br />
manifolds!), called Stratified Morse Theory. After going through the<br />
appropriate definitions and briefly reviewing the results of classical<br />
Morse Theory, we'll discuss the so-called Main Theorem of Stratified<br />
Morse Theory and survey some of its consequences.<br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Birational geometry: existence of rational curves<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight. <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=20125Graduate Algebraic Geometry Seminar2020-10-13T16:27:43Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:25pm<br />
<br />
'''Where:''' Van Vleck B317<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
== Fall 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#E0E0E0"| September 30<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 30| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 5<br />
| bgcolor="#C6D46E"| Yifan Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 5| On the Analytic Side (GAGA)]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 14<br />
| bgcolor="#C6D46E"| Owen Goff<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 14| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 21<br />
| bgcolor="#C6D46E"| Roufan Jiang<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 21| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 28<br />
| bgcolor="#C6D46E"| Erika Pirnes<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 28| Introduction to representation theory via an example]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 4<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 4| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 11<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 11| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 18<br />
| bgcolor="#C6D46E"| Yunfan He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 18| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| November November 25<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 2<br />
| bgcolor="#C6D46E"| Peter Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 2| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 9<br />
| bgcolor="#C6D46E"| Wendy Cheng<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 9| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 16<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 16| TBD]]<br />
|}<br />
</center><br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Birational geometry: existence of rational curves]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.<br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.<br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Stanley-Reisner Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.<br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Early on in the semester, Colin told us a bit about Morse<br />
Theory, and how it lets us get a handle on the (classical) topology of<br />
smooth complex varieties. As we all know, however, not everything in<br />
life goes smoothly, and so too in algebraic geometry. Singular<br />
varieties, when given the classical topology, are not manifolds, but<br />
they can be described in terms of manifolds by means of something called<br />
a Whitney stratification. This allows us to develop a version of Morse<br />
Theory that applies to singular spaces (and also, with a bit of work, to<br />
smooth spaces that fail to be nice in other ways, like non-compact<br />
manifolds!), called Stratified Morse Theory. After going through the<br />
appropriate definitions and briefly reviewing the results of classical<br />
Morse Theory, we'll discuss the so-called Main Theorem of Stratified<br />
Morse Theory and survey some of its consequences.<br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Birational geometry: existence of rational curves<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight. <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=19368Graduate Algebraic Geometry Seminar2020-04-21T19:09:54Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:25pm<br />
<br />
'''Where:''' Van Vleck B317<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Birational geometry: existence of rational curves]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.<br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.<br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Stanley-Reisner Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.<br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Early on in the semester, Colin told us a bit about Morse<br />
Theory, and how it lets us get a handle on the (classical) topology of<br />
smooth complex varieties. As we all know, however, not everything in<br />
life goes smoothly, and so too in algebraic geometry. Singular<br />
varieties, when given the classical topology, are not manifolds, but<br />
they can be described in terms of manifolds by means of something called<br />
a Whitney stratification. This allows us to develop a version of Morse<br />
Theory that applies to singular spaces (and also, with a bit of work, to<br />
smooth spaces that fail to be nice in other ways, like non-compact<br />
manifolds!), called Stratified Morse Theory. After going through the<br />
appropriate definitions and briefly reviewing the results of classical<br />
Morse Theory, we'll discuss the so-called Main Theorem of Stratified<br />
Morse Theory and survey some of its consequences.<br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Birational geometry: existence of rational curves<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight. <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=19355Graduate Algebraic Geometry Seminar2020-04-14T16:31:31Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:25pm<br />
<br />
'''Where:''' Van Vleck B317<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.<br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.<br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Stanley-Reisner Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.<br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Early on in the semester, Colin told us a bit about Morse<br />
Theory, and how it lets us get a handle on the (classical) topology of<br />
smooth complex varieties. As we all know, however, not everything in<br />
life goes smoothly, and so too in algebraic geometry. Singular<br />
varieties, when given the classical topology, are not manifolds, but<br />
they can be described in terms of manifolds by means of something called<br />
a Whitney stratification. This allows us to develop a version of Morse<br />
Theory that applies to singular spaces (and also, with a bit of work, to<br />
smooth spaces that fail to be nice in other ways, like non-compact<br />
manifolds!), called Stratified Morse Theory. After going through the<br />
appropriate definitions and briefly reviewing the results of classical<br />
Morse Theory, we'll discuss the so-called Main Theorem of Stratified<br />
Morse Theory and survey some of its consequences.<br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=19184Graduate Algebraic Geometry Seminar2020-03-03T05:04:53Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:25pm<br />
<br />
'''Where:''' Van Vleck B317<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.<br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.<br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Mine'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=19183Graduate Algebraic Geometry Seminar2020-03-03T05:04:08Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:25pm<br />
<br />
'''Where:''' Van Vleck B317<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.<br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.<br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Mine'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=19057Graduate Algebraic Geometry Seminar2020-02-18T20:51:39Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:25pm<br />
<br />
'''Where:''' Van Vleck B317<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Mine'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=19056Graduate Algebraic Geometry Seminar2020-02-18T20:50:07Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:25pm<br />
<br />
'''Where:''' Van Vleck B317<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Mine'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=19055Graduate Algebraic Geometry Seminar2020-02-18T20:48:52Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:25pm<br />
<br />
'''Where:''' Van Vleck B317<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.)<br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Mine'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Madison_Math_Circle&diff=19052Madison Math Circle2020-02-18T16:37:37Z<p>Crowley: </p>
<hr />
<div>[[Image:logo.png|right|440px]]<br />
<br />
For the site in Spanish, visit [[Math Circle de Madison]]<br />
=What is a Math Circle?=<br />
The Madison Math Circle is a weekly series of mathematically based activities aimed at interested middle school and high school students. It is an outreach program organized by the UW Math Department. Our goal is to provide a taste of exciting ideas in math and science. In the past we've had talks about plasma and weather in outer space, video game graphics, and encryption. In the sessions, students (and parents) are often asked to explore problems on their own, with the presenter facilitating a discussion. The talks are independent of one another, so new students are welcome at any point.<br />
<br />
The level of the audience varies quite widely, including a mix of middle school and high school students, and the speakers generally address this by considering subjects that will be interesting for a wide range of students.<br />
<br />
<br />
[[Image: MathCircle_2.jpg|500px]] [[Image: MathCircle_4.jpg|500px]] <br />
<br />
<br />
After each talk we'll have pizza provided by the Mathematics Department, and students will have an opportunity to mingle and chat with the speaker and with other participants, to ask questions about some of the topics that have been discussed, and also about college, careers in science, etc.<br />
<br />
'''The Madison Math circle was featured in Wisconsin State Journal:''' [http://host.madison.com/wsj/news/local/education/local_schools/school-spotlight-madison-math-circle-gives-young-students-a-taste/article_77f5c042-0b3d-11e1-ba5f-001cc4c03286.html check it out]!<br />
<br />
=All right, I want to come!=<br />
<br />
We have a weekly meeting, <b>Monday at 6pm in 3255 Helen C White Library</b>, during the school year. <b>New students are welcome at any point! </b> There is no fee and the talks are independent of one another, so you can just show up any week, but we ask all participants to take a moment to register by following the link below:<br />
<br />
[https://uwmadison.co1.qualtrics.com/jfe/form/SV_e9WdAs2SXNurWFD '''Math Circle Registration Form''']<br />
<br />
All of your information is kept private, and is only used by the Madison Math Circle organizer to help run the Circle. <br />
<br />
If you are a student, we hope you will tell other interested students about these talks, and speak with your parents or with your teacher about organizing a car pool to the UW campus. If you are a parent or a teacher, we hope you'll tell your students about these talks and organize a car pool to the UW (all talks take place in 3255 Helen C White Library, on the UW-Madison campus, right next to the Memorial Union).<br />
<br />
<br />
==Directions and parking==<br />
Our meetings are held on the 3rd floor of Helen C. White Hall in room 3255.<br />
<br />
<div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><br />
[[File: Helencwhitemap.png|400px]]</div><br />
<br />
'''Parking.''' Parking on campus is rather limited. Here is as list of some options:<br />
<br />
*There is a parking garage in the basement of Helen C. White, with an hourly rate. Enter from Park Street.<br />
*A 0.5 mile walk to Helen C. White Hall via [http://goo.gl/cxTzJY these directions], many spots ('''free starting 4:30pm''') [http://goo.gl/maps/Gkx1C in Lot 26 along Observatory Drive].<br />
*A 0.3 mile walk to Helen C. White Hall via [http://goo.gl/yMJIRd these directions], many spots ('''free starting 4:30pm''') [http://goo.gl/maps/vs17X in Lot 34]. <br />
*A 0.3 mile walk to Helen C. White Hall via [http://goo.gl/yMJIRd these directions], 2 metered spots (25 minute max) [http://goo.gl/maps/ukTcu in front of Lathrop Hall].<br />
*A 0.2 mile walk to Helen C. White Hall via [http://goo.gl/b8pdk2 these directions] 6 metered spots (25 minute max) around [http://goo.gl/maps/6EAnc the loop in front of Chadbourne Hall] .<br />
*For more information, see the [http://transportation.wisc.edu/parking/parking.aspx UW-Madison Parking Info website].<br />
<br />
==Email list==<br />
The best way to keep up to date with the what is going is by signing up for our email list. Send an empty email to join-mathcircle@lists.wisc.edu<br />
<br />
==Contact the organizers==<br />
The Madison Math Circle is organized by a group of professors and graduate students from the [http://www.math.wisc.edu Department of Mathematics] at the UW-Madison. If you have any questions, suggestions for topics, or so on, just email the '''organizers''' [mailto:cbooms@wisc.edu here]. We are always interested in feedback!<br />
<center><br />
<gallery widths=500px heights=250px mode="packed"><br />
File:de.jpg|[https://www.math.wisc.edu/~derman/ Prof. Daniel Erman]<br />
<!--File:Betsy.jpg|[http://www.math.wisc.edu/~stovall/ Prof. Betsy Stovall]--><br />
</gallery><br />
<br />
<gallery widths=500px heights=250px mode="packed"><br />
<!--File:juliettebruce.jpg|[http://www.math.wisc.edu/~juliettebruce/ Juliette Bruce] File:Ee.jpg|[http://www.math.wisc.edu/~evaelduque/ Eva Elduque] File:mrjulian.jpg|[http://www.math.wisc.edu/~mrjulian/ Ryan Julian] File:soumyasankar.jpg|[http://www.math.wisc.edu/~soumyasankar Soumya Sankar]--><br />
File:caitlynbooms.jpg|[https://sites.google.com/wisc.edu/cbooms Caitlyn Booms]<br />
File:colincrowley.jpg|[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
File:hyunjongkim.jpg|Hyun Jong Kim<br />
File:Xshen.jpg|[https://www.math.wisc.edu/~xshen// Xiao Shen]<br />
File:connorsimpson.jpg|[http://www.math.wisc.edu/~csimpson6/ Connor Simpson]<br />
</gallery><br />
</center><br />
<br />
==Donations==<br />
Please consider donating to the Madison Math Circle. As noted in our [https://www.math.wisc.edu/wiki/images/Math_Circle_Newsletter.pdf annual report], our main costs consist of pizza and occasional supplies for the speakers. So far our costs have been covered by donations from the UW Mathematics Department as well as a generous gifts from a private donor. But our costs are rising, primarily because this year we expect to hold more meetings than in any previous year. In fact, this year, we expect to spend at least $2500 on pizza and supplies alone.<br />
<br />
So please consider donating to support your math circle! The easiest way to donate is to go to the link:<br />
<br />
[http://www.math.wisc.edu/donate Online Donation Link]<br />
<br />
There are instructions on that page for donating to the Math Department. <b> Be sure and add a Gift Note saying that the donation is intended for the "Madison Math Circle"!</b> The money goes into the Mathematics Department Annual Fund and is routed through the University of Wisconsin Foundation, which is convenient for record-keeping, etc.<br />
<br />
Alternately, you can bring a check to one of the Math Circle Meetings. If you write a check, be sure to make it payable to the "WFAA" and add the note "Math Circle Donation" on the check. <br />
<br />
Or you can just pay in cash, and we'll give you a receipt.<br />
<br />
==Help us grow!==<br />
If you like Math Circle, please help us continue to grow! Students, parents, and teachers can help by:<br />
* Like our [https://facebook.com/madisonmathcircle '''Facebook Page'''] and share our events with others! <br />
* Posting our [https://www.math.wisc.edu/wiki/images/MMC_Flyer_2016.pdf '''flyer'''] at schools or anywhere that might have interested students.<br />
* Discussing the Math Circle with students, parents, teachers, administrators, and others.<br />
* Making an announcement about Math Circle at PTO meetings.<br />
* Donating to Math Circle.<br />
Contact the organizers if you have questions or your own ideas about how to help out.<br />
<br />
=Meetings for Fall 2019=<br />
<br />
<center><br />
<br />
Talks start at '''6pm in room 3255 of Helen C. White Library''', unless otherwise noted.<br />
<br />
</center><br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="1" cellpadding="14" cellspacing="0"<br />
|-<br />
! colspan="3" style="background: #e8b2b2;" align="center" | Fall 2019<br />
|-<br />
! Date !! Speaker !! Topic<br />
|-<br />
| September 23, 2019 || Soumya Sankar || Why don't map makers like high heels?<br />
|-<br />
| September 30, 2019 || Erika Pirnes || Why do ice hockey players fall in love with mathematicians?<br />
|-<br />
| October 7, 2019 || Uri Andrews || Self-reference, proofs, and computer programming<br />
|-<br />
| October 14, 2019 || James Hanson || When is a puzzle impossible?<br />
|-<br />
| October 21, 2019 || Owen Goff || Symbolic Logic and How It's Really Just Arithmetic<br />
|-<br />
| October 28, 2019 || Ian Seong || Counting, but Not Like Kindergarteners<br />
|-<br />
| November 4, 2019 || Omer Mermelstein || Ciphers: To Gibberish and Back Again<br />
|-<br />
| November 11, 2019 || Colin Crowley || Many Pennies<br />
|-<br />
| November 18, 2019 || Daniel Corey || The K<span>&#246;</span>nigsberg Bridge Problem<br />
|-<br />
|}<br />
<br />
</center><br />
<br />
=Meetings for Spring 2020=<br />
<br />
<center><br />
<br />
Talks start at '''6pm in room 3255 of Helen C. White Library''', unless otherwise noted.<br />
<br />
</center><br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="1" cellpadding="14" cellspacing="0"<br />
|-<br />
! colspan="3" style="background: #e8b2b2;" align="center" | Spring 2020<br />
|-<br />
! Date !! Speaker !! Topic<br />
|-<br />
| January 27, 2020 || Caitlyn Booms || Magic or Math?<br />
|-<br />
| February 3, 2020 || Erika Pirnes || Finding Your Roots<br />
|-<br />
| February 10, 2020 || Xiao Shen || Constructing the 17-gon<br />
|-<br />
| February 17, 2020 || Ben Bruce || 1+1=2 and Other Integer Partitions<br />
|-<br />
| February 24, 2020 || Brandon Boggess || TBD<br />
|-<br />
| March 2, 2020 || Solly Parenti || TBD<br />
|-<br />
| March 9, 2020 || Connor Simpson || Counting ways to color graphs<br />
|-<br />
| March 23, 2020 || TBD || TBD<br />
|-<br />
| March 30, 2020 || TBD || TBD<br />
|-<br />
| April 6, 2020 || Ivan Aidun || TBD<br />
|-<br />
| April 13, 2020 || TBD || TBD<br />
|-<br />
| April 20, 2020 || TBD || TBD<br />
|-<br />
|}<br />
<br />
</center><br />
<br />
=Off-Site Meetings=<br />
<br />
We will hold some Math Circle meetings at local high schools on early release days. If you are interesting in having us come to your high school, please contact us!<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="1" cellpadding="14" cellspacing="0"<br />
|-<br />
! colspan="5" style="background: #e8b2b2;" align="center" | Fall 2019<br />
|-<br />
|-<br />
! Date !! Location !! Speaker !! Title !! Abstract<br />
|-<br />
| October 7, 2019 || 2:45pm East High || Solly Parenti || Tangled Up in Two || Every tangled cord you have ever encountered is secretly a number. Once you learn how to count these cords, cleaning your room will be as easy as 1-2-3.<br />
|-<br />
| November 4, 2019 || 2:45pm James Madison Memorial || Caitlyn Booms || Sneaky Segments || We call a line segment drawn between two lattice points in the coordinate plane sneaky if it does not pass through any other lattice points. During this presentation, we will try to understand exactly when this happens, and we'll discuss how to calculate the probability that two randomly chosen lattice points are connected by a sneaky segment.<br />
|-<br />
| November 11, 2019 || 2:45pm East High || Maya Banks || Tic-Tac-Topology || Tic-Tac-Toe is a game usually played on a flat piece of paper. In this standard setting, there is winning strategy--that is, if the player who goes first chooses their moves correctly, they will never lose. But we can also play Tic-Tac-Toe on a surface that isn't lying flat in a plane! In this talk, we will explore the game of Tic-Tac-Toe on cylinders, donuts, and even some wilder surfaces. We'll look for optimal strategies, and learn some topology in the process.<br />
|-<br />
| December 16, 2019 || 2:45pm James Madison Memorial || Daniel Erman || Really Big Numbers || We will discuss the role that really really, really big numbers play in modern mathematics and in science. This will be a discussion of estimation and an introduction to some of the ways that mathematicians express unfathomably big numbers.<br />
|}<br />
<br />
<br />
<br />
{| style="color:black; font-size:120%" border="1" cellpadding="14" cellspacing="0"<br />
|-<br />
! colspan="5" style="background: #e8b2b2;" align="center" | Spring 2020<br />
|-<br />
|-<br />
! Date !! Location !! Speaker !! Title !! Abstract<br />
|-<br />
| February 17, 2020 || 2:45pm James Madison Memorial || Maya Banks || Tic-Tac-Topology || Tic-Tac-Toe is a game usually played on a flat piece of paper. In this standard setting, there is winning strategy--that is, if the player who goes first chooses their moves correctly, they will never lose. But we can also play Tic-Tac-Toe on a surface that isn't lying flat in a plane! In this talk, we will explore the game of Tic-Tac-Toe on cylinders, donuts, and even some wilder surfaces. We'll look for optimal strategies, and learn some topology in the process.<br />
|-<br />
| March 9, 2020 || 2:45pm East High || Michel Alexis || TBD || TBD<br />
|-<br />
| April 13, 2020 || 2:45pm James Madison Memorial || Juliette Bruce || TBD || TBD<br />
|-<br />
| April 20, 2020 || 2:45pm East High || TBD || TBD || TBD<br />
|}<br />
</center><br />
<br />
=Useful Resources=<br />
==Annual Reports==<br />
[https://www.math.wisc.edu/wiki/images/Math_Circle_Newsletter.pdf 2013-2014 Annual Report]<br />
<br />
== Archived Abstracts ==<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_2016-2017 2016 - 2017 Math Circle Page]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts_2016-2017 2016 - 2017 Abstracts]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_2015-2016 2015 - 2016 Math Circle Page]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Math_Circle_de_Madison_2015-2016 2015 - 2016 Math Circle Page (Spanish)]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts_2015-2016 2015 - 2015 Abstracts]<br />
<br />
[[Archived Math Circle Material]]<br />
<br />
==Link for presenters (in progress)==<br />
[https://www.math.wisc.edu/wiki/index.php/Math_Circle_Presentations Advice For Math Circle Presenters]<br />
<br />
[http://www.geometer.org/mathcircles/ Sample Talk Ideas/Problems from Tom Davis]<br />
<br />
[https://www.mathcircles.org/activities Sample Talks from the National Association of Math Circles]<br />
<br />
[https://epdf.pub/circle-in-a-box715623b97664e247f2118ddf7bec4bfa35437.html "Circle in a Box"]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Matroids_seminar&diff=17562Matroids seminar2019-07-18T22:47:15Z<p>Crowley: </p>
<hr />
<div>The matroids seminar & reading group meets '''10:00--10:45 on Fridays in Van Vleck 901''' in order to discuss matroids from a variety of viewpoints.<br />
In particular, we aim to<br />
* survey open conjectures and recent work in the area<br />
* compute many interesting examples<br />
* discover concrete applications<br />
For updates, join our mailing list, matroids [at] lists.wisc.edu<br />
<br />
We are happy to have new leaders of the discussion, and the wide range of topics to which matroids are related mean that each week is a great chance for a new participant to drop in! If you would like to talk but need ideas, see the [[Matroids seminar/ideas]] page.<br />
<br />
To help develop an inclusive environment, a subset of the organizers will be available before the talk in the ninth floor lounge to informally discuss background material e.g., "What is a variety?", "What is a circuit?", "What is a greedy algorithm?" (this is especially for those coming from an outside area).<br />
<br />
'''Organizers''': Colin Crowley, Connor Simpson; Daniel Corey, Jose Israel Rodriguez<br />
{|cellpadding="8px" style="vertical-align:top; text-align:left; cellpadding:10px;"<br />
|-<br />
|5/03/2019<br />
|<br />
<div style="font-weight:bold;">[http://www.math.wisc.edu/~csimpson6/ Connor Simpson]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>A flip-free proof of the Heron-Rota-Welsh conjecture</i></div><br />
<div class="mw-collapsible-content"><br />
The simplicial presentation of a matroid yields a flip-free proof of the Kahler package in degree 1 for the Chow ring of a matroid, which is enough to give a new proof of the Heron-Rota-Welsh conjecture.<br />
This talk is more or less a continuation of the one that Chris Eur gave earlier in the semester in the algebra seminar, and is based on the same joint work with Spencer Backman and Chris Eur.<br />
</div></div><br />
|-<br />
|4/12/2019 & 4/19/2019<br />
|<br />
<div style="font-weight:bold;">No seminar</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Many organizers are traveling.</i></div><br />
<div class="mw-collapsible-content"><br />
None. <br />
</div></div><br />
|-<br />
|4/05/2019<br />
|<br />
<div style="font-weight:bold;">Jose Israel Rodriguez</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Planar pentads, polynomial systems, and polymatroids</i></div><br />
<div class="mw-collapsible-content"><br />
Computing exceptional sets using fiber products naturally yields multihomogeneous systems of polynomial equations. <br />
In this talk, I will utilize a variety of tools from the forthcoming paper "A numerical toolkit for multiprojective varieties" to work out an example from kinematics: exceptional planar pentads.<br />
In particular, we will derive a multihomogeneous polynomial system whose solutions have meaning in kinematics and discuss how polymatroids play a role in describing the solutions. <br />
</div></div><br />
|-<br />
|3/29/2019<br />
|<br />
<div style="font-weight:bold;">[https://sites.google.com/view/colincrowley/home Colin Crowley]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Binary matroids and Seymour's decomposition in coding theory</i></div><br />
<div class="mw-collapsible-content"><br />
We will begin by discussing the equivalence between a binary matroid and a binary linear code. And then following [https://ac.els-cdn.com/009589568990052X/1-s2.0-009589568990052X-main.pdf?_tid=a1d8598e-c5f2-4c07-8d7b-0843e88c416f&acdnat=1552505468_33915774714b2e407e4fd7001ba33100 this paper] and [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4544972 this one], we will describe the Maximum Likelihood decoding problem and then outline how Seymour's decomposition theorem for regular matroids led to a polynomial time algorithm on a subclass of binary linear codes.<br />
</div></div><br />
|-<br />
|3/15/2019<br />
|<br />
<div style="font-weight:bold;">[https://sites.google.com/site/dcorey2814/home Daniel Corey]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>The geometry of thin Schubert cells</i></div><br />
<div class="mw-collapsible-content"><br />
We will cover the distinction between the thin Schubert cell of a matroid and the realization space of a matroid, how to compute examples, Mnev universality, and time permitting, maps between thin Schubert cells. <br />
</div></div><br />
|-<br />
|3/8/2019<br />
|<br />
<div style="font-weight:bold;">Vladmir Sotirov</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>is sick</i></div><br />
<div class="mw-collapsible-content"><br />
Plague and pestilence!<br />
</div></div><br />
|-<br />
|3/1/2019<br />
|<br />
<div style="font-weight:bold;">[https://math.berkeley.edu/~ceur/ Chris Eur]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>The multivariate Tutte polynomial of a flag matroid</i></div><br />
<div class="mw-collapsible-content"><br />
Flag matroids are combinatorial objects whose relation to ordinary matroids are akin to that of flag varieties to Grassmannians. We define a multivariate Tutte polynomial of a flag matroid, and show that it is Lorentzian in the sense of [https://arxiv.org/abs/1902.03719 Branden-Huh '19]. As a consequence, we obtain a flag matroid generalization of Mason’s conjecture concerning the f-vector of independent subsets of a matroid. This is an on-going joint work with June Huh.<br />
</div></div><br />
|-<br />
|2/22/2019<br />
|<br />
<div style="font-weight:bold;">[https://www.math.wisc.edu/~wang/ Botong Wang]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>The Kazhdan-Lusztig polynomial of a matroid</i></div><br />
<div class="mw-collapsible-content"><br />
Classically, Kazdhan-Lusztig polynomials are associated to intervals of the Bruhat poset of a Coxeter group. We will discuss an analogue of Kazdhan-Lusztig polynomials for matroids, including results and conjectures from [https://arxiv.org/pdf/1611.07474.pdf these] [https://arxiv.org/pdf/1412.7408.pdf two] papers. <br />
</div></div><br />
|-<br />
|2/15/2019<br />
|<br />
<div style="font-weight:bold;">[https://sites.google.com/view/colincrowley/home Colin Crowley]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Matroid polytopes</i></div><br />
<div class="mw-collapsible-content"><br />
We outline the original formulation of matroid polytopes as moment polytopes of subvarieties of the Grassmanian, following [http://www.math.ias.edu/~goresky/pdf/combinatorial.jour.pdf Combinatorial Geometries, Convex Polyhedra, and Schbert Cells].<br />
</div></div><br />
|-<br />
|2/8/2019<br />
|<br />
<div style="font-weight:bold;">[http://www.math.wisc.edu/~csimpson6/ Connor Simpson]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Proving the Heron-Rota-Welsh conjecture</i></div><br />
<div class="mw-collapsible-content"><br />
We outline the proof of the Heron-Rota-Welsh conjecture given by Adiprasito, Huh, and Katz in their paper [https://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries]<br />
</div></div><br />
|-<br />
|1/25/2019 & 2/1/2019<br />
|<br />
<div style="font-weight:bold;">[https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Algebraic matroids in action</i></div><br />
<div class="mw-collapsible-content"><br />
We discuss algebraic matroids and their applications; see [https://arxiv.org/abs/1809.00865 Algebraic Matroids in Action].<br />
</div></div><br />
|-<br />
|1/18/2019<br />
|<br />
<div style="font-weight:bold;">[https://sites.google.com/site/dcorey2814/home Daniel Corey]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Introduction to matroids</i></div><br />
<div class="mw-collapsible-content"><br />
We'll cover the basic definitions and some examples, roughly following [http://web.ma.utexas.edu/users/sampayne/pdf/Math648Lecture3.pdf these notes].<br />
</div></div><br />
|-<br />
|}</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Curl_Summer_2019&diff=17446Curl Summer 20192019-05-16T16:56:05Z<p>Crowley: </p>
<hr />
<div>This summer there are two students participating in CURL, each doing a project in applied algebraic geometry.<br />
<br />
We will meet from Monday, May 13th through Friday, June 28th on weekdays from 9:00 to 12:00.<br />
<br />
'''Students:''' Jacob Weiker, Jacob Zoromski<br />
<br />
'''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
===Computing Mixed Nash Equilibria===<br />
'''Student:''' Jacob Weiker <br />
<br />
Problems in economics are often modeled as multiplayer games. In a model where each player chooses between multiple strategies according to a probability distribution, of interest are the so called Mixed Nash Equilibria: situations where no player can benefit by changing their probability of using a particular strategy. In this project we will cover background material on solving polynomial systems, and then study the polynomial formulation of various Mixed Nash Equilibria problems. <br />
<br />
===Computational algebraic geometry for Algebraic Kinematics ===<br />
'''Student:''' Jacob Zoromski <br />
<br />
Various problems in Kinematics lead to polynomial systems. We will cover background on solving these systems, and then compute examples involving exceptional mechanisms and transitional mechanisms.<br />
<br />
In both projects we will focus on using software to compute many examples, and forming observations and conjectures about the real solutions. With remaining time we aim to prove theoretical results or develop implementations.</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Curl_Summer_2019&diff=17445Curl Summer 20192019-05-16T16:55:43Z<p>Crowley: </p>
<hr />
<div>This summer there are two students participating in CURL, each doing a project in applied algebraic geometry.<br />
<br />
We will meet from Monday, May 13th through Friday, June 28th on weekdays from 9:00 to 12:00.<br />
<br />
'''Students: Jacob Weiker, Jacob Zoromski'''<br />
<br />
'''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
===Computing Mixed Nash Equilibria===<br />
'''Student:''' Jacob Weiker <br />
<br />
Problems in economics are often modeled as multiplayer games. In a model where each player chooses between multiple strategies according to a probability distribution, of interest are the so called Mixed Nash Equilibria: situations where no player can benefit by changing their probability of using a particular strategy. In this project we will cover background material on solving polynomial systems, and then study the polynomial formulation of various Mixed Nash Equilibria problems. <br />
<br />
===Computational algebraic geometry for Algebraic Kinematics ===<br />
'''Student:''' Jacob Zoromski <br />
<br />
Various problems in Kinematics lead to polynomial systems. We will cover background on solving these systems, and then compute examples involving exceptional mechanisms and transitional mechanisms.<br />
<br />
In both projects we will focus on using software to compute many examples, and forming observations and conjectures about the real solutions. With remaining time we aim to prove theoretical results or develop implementations.</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Curl_Summer_2019&diff=17444Curl Summer 20192019-05-16T16:54:10Z<p>Crowley: </p>
<hr />
<div>This summer there are two students participating in CURL, each doing a project in applied algebraic geometry.<br />
<br />
We will meet from Monday, May 13th through Friday, June 28th on weekdays from 9:00 to 12:00.<br />
<br />
===Computing Mixed Nash Equilibria===<br />
'''Student:''' Jacob Weiker<br />
'''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
Problems in economics are often modeled as multiplayer games. In a model where each player chooses between multiple strategies according to a probability distribution, of interest are the so called Mixed Nash Equilibria: situations where no player can benefit by changing their probability of using a particular strategy. In this project we will cover background material on solving polynomial systems, and then study the polynomial formulation of various Mixed Nash Equilibria problems. <br />
<br />
===Computational algebraic geometry for Algebraic Kinematics ===<br />
'''Student:''' Jacob Zoromski '''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
Various problems in Kinematics lead to polynomial systems. We will cover background on solving these systems, and then compute examples involving exceptional mechanisms and transitional mechanisms.<br />
<br />
In both projects we will focus on using software to compute many examples, and forming observations and conjectures about the real solutions. With remaining time we aim to prove theoretical results or develop implementations.</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Curl_Summer_2019&diff=17443Curl Summer 20192019-05-16T16:51:47Z<p>Crowley: </p>
<hr />
<div>This summer there are two students participating in CURL, each doing a project in applied algebraic geometry.<br />
<br />
===Computing Mixed Nash Equilibria===<br />
'''Student:''' Jacob Weiker<br />
'''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
Problems in economics are often modeled as multiplayer games. In a model where each player chooses between multiple strategies according to a probability distribution, of interest are the so called Mixed Nash Equilibria: situations where no player can benefit by changing their probability of using a particular strategy. In this project we will cover background material on solving polynomial systems, and then study the polynomial formulation of various Mixed Nash Equilibria problems. <br />
<br />
===Computational algebraic geometry for Algebraic Kinematics ===<br />
'''Student:''' Jacob Zoromski '''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
Various problems in Kinematics lead to polynomial systems. We will cover background on solving these systems, and then compute examples involving exceptional mechanisms and transitional mechanisms.<br />
<br />
In both projects we will focus on using software to compute many examples, and forming observations and conjectures about the real solutions. With remaining time we aim to prove theoretical results or develop implementations.</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Curl_Summer_2019&diff=17442Curl Summer 20192019-05-16T16:50:02Z<p>Crowley: /* Computational algebraic geometry for Algebraic Kinematics */</p>
<hr />
<div>This summer there are two students participating in CURL, each doing a project in applied algebraic geometry.<br />
<br />
===Computing Mixed Nash Equilibria===<br />
'''Student:''' Jacob Weiker<br />
'''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
Problems in economics are often modeled as multiplayer games. In a model where each player chooses between multiple strategies according to a probability distribution, of interest are the so called Mixed Nash Equilibria: situations where no player can benefit by changing their probability of using a particular strategy. In this project we will cover background material on solving polynomial systems, and then study the polynomial formulation of various Mixed Nash Equilibria problems. We will focus on using software to compute many examples, and forming observations and conjectures about the real solutions. With remaining time we aim to prove theoretical results or develop implementations.<br />
<br />
===Computational algebraic geometry for Algebraic Kinematics ===<br />
'''Student:''' Jacob Zoromski '''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
Various problems in Kinematics lead to polynomial systems. We will cover background on solving these systems, and then compute examples involving exceptional mechanisms and transitional mechanisms.</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Curl_Summer_2019&diff=17441Curl Summer 20192019-05-16T15:35:52Z<p>Crowley: /* Finding Exceptional Mechanisms in Kinematics */</p>
<hr />
<div>This summer there are two students participating in CURL, each doing a project in applied algebraic geometry.<br />
<br />
===Computing Mixed Nash Equilibria===<br />
'''Student:''' Jacob Weiker<br />
'''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
Problems in economics are often modeled as multiplayer games. In a model where each player chooses between multiple strategies according to a probability distribution, of interest are the so called Mixed Nash Equilibria: situations where no player can benefit by changing their probability of using a particular strategy. In this project we will cover background material on solving polynomial systems, and then study the polynomial formulation of various Mixed Nash Equilibria problems. We will focus on using software to compute many examples, and forming observations and conjectures about the real solutions. With remaining time we aim to prove theoretical results or develop implementations.<br />
<br />
===Computational algebraic geometry for Algebraic Kinematics ===<br />
'''Student:''' Jacob Zoromski '''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Curl_Summer_2019&diff=17440Curl Summer 20192019-05-16T15:31:17Z<p>Crowley: /* Computing Mixed Nash Equilibria */</p>
<hr />
<div>This summer there are two students participating in CURL, each doing a project in applied algebraic geometry.<br />
<br />
===Computing Mixed Nash Equilibria===<br />
'''Student:''' Jacob Weiker<br />
'''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
Problems in economics are often modeled as multiplayer games. In a model where each player chooses between multiple strategies according to a probability distribution, of interest are the so called Mixed Nash Equilibria: situations where no player can benefit by changing their probability of using a particular strategy. In this project we will cover background material on solving polynomial systems, and then study the polynomial formulation of various Mixed Nash Equilibria problems. We will focus on using software to compute many examples, and forming observations and conjectures about the real solutions. With remaining time we aim to prove theoretical results or develop implementations.<br />
<br />
===Finding Exceptional Mechanisms in Kinematics===</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Curl_Summer_2019&diff=17439Curl Summer 20192019-05-16T15:13:37Z<p>Crowley: /* Computing Mixed Nash Equilibria */</p>
<hr />
<div>This summer there are two students participating in CURL, each doing a project in applied algebraic geometry.<br />
<br />
===Computing Mixed Nash Equilibria===<br />
'''Student:''' Jacob Weiker<br />
'''Mentors:''' [https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez], Colin Crowley<br />
<br />
===Finding Exceptional Mechanisms in Kinematics===</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Curl_Summer_2019&diff=17438Curl Summer 20192019-05-16T15:10:17Z<p>Crowley: Created page with "This summer there are two students participating in CURL, each doing a project in applied algebraic geometry. ===Computing Mixed Nash Equilibria=== ===Finding Exceptional Me..."</p>
<hr />
<div>This summer there are two students participating in CURL, each doing a project in applied algebraic geometry.<br />
<br />
===Computing Mixed Nash Equilibria===<br />
<br />
===Finding Exceptional Mechanisms in Kinematics===</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Summer_2019_Algebraic_Geometry_Reading_Group&diff=17374Summer 2019 Algebraic Geometry Reading Group2019-04-24T03:00:28Z<p>Crowley: </p>
<hr />
<div>This is the page for the Summer 2019 Algebraic Geometry Reading Group. <br />
<br />
== Resources ==<br />
<br />
We plan to primarily use the newest version of Ravi Vakil's The Rising Sea: Foundations of Algebraic Geometry, which can be found here: http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf.<br />
<br />
At times we may also use Hartshorne's Algebraic Geometry.<br />
<br />
== Schedule ==<br />
<br />
10 Total Weeks: May 13-31, June 24-August 9<br />
<br />
Roughly 3 meetings per week for 1-1.5 hours each.<br />
<br />
Exact days will be determined based on the schedules of the participants.<br />
<br />
<br />
'''Optimistic reading schedule (all chapters from Vakil):'''<br />
<br />
'''Week of May 13:''' Ch. 3, Ch. 4 (Affine schemes, structure sheaf)<br />
<br />
'''Week of May 20:''' Ch. 5, Start Ch. 6 (Properties of schemes, morphisms of schemes)<br />
<br />
'''Week of May 27:''' Finish Ch. 6, Ch. 7 (Classes of morphisms of schemes)<br />
<br />
'''Week of June 24:''' Ch. 8, Start Ch. 9 (Closed embeddings and Cartier divisors, fibered product of schemes)<br />
<br />
'''Week of July 1:''' Finish Ch. 9, Ch. 10 (Separated and proper morphisms, varieties)<br />
<br />
'''Week of July 8:''' Ch. 11, Start Ch. 12 (Dimension, regularity and smoothness)<br />
<br />
'''Week of July 15:''' Finish Ch. 12, Ch. 13 (Quasicoherent and coherent sheaves)<br />
<br />
'''Week of July 22:''' Ch. 14, Start Ch. 15 (Line bundles, projective schemes)<br />
<br />
'''Week of July 29:''' Finish Ch. 15, Ch. 16 (Pushforwards and pullbacks of quasicoherent sheaves)<br />
<br />
'''Week of August 5:''' Ch. 18, Ch. 19, Ch. 21 (Cech cohomology, curves, differentials)<br />
<br />
== General Meeting Structure ==<br />
<br />
This reading seminar will be structured as follows. Every meeting will have an assigned "leader," who will usually be one of the reading group participants, but could at times be an older grad student or professor. It will be expected that everyone attending will read the assigned chapters prior to the meeting. The "leader" is expected to additionally work out some examples prior and will be responsible for guiding the group discussion during the meeting. Meetings will primarily be spent discussing questions that everyone has about the reading and going through examples together. Depending on the interest of the group, we may also have problem solving sessions.<br />
<br />
'''If you are interested in joining this reading group, please contact Caitlyn Booms at cbooms@wisc.edu by May 8, 2019 and join the mailing list by emailing join-ag (at) lists.wisc.edu'''<br />
<br />
== Summer plans ==<br />
<br />
If you feel like telling us your general plans for the summer, so that we'll know when you are around Madison, please do so here:<br />
<br />
Caitlyn: out of town May 29-June 23 and Aug. 17-25<br />
<br />
Colin: Out of town July 1-15 and one week sometime in June probably</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=17248Graduate Algebraic Geometry Seminar2019-03-31T00:14:33Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:25pm<br />
<br />
'''Where:''' Van Vleck B317 (Spring 2019)<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cbooms@wisc.edu Caitlyn] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
<br />
==Ed Dewey's Wish List Of Olde==<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Famous Theorems===<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
__NOTOC__<br />
<br />
== Spring 2019 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| February 6<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[#February 6| Heisenberg Groups and the Fourier Transform]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 13<br />
| bgcolor="#C6D46E"| David Wagner<br />
| bgcolor="#BCE2FE"|[[#February 13| DG potpourri]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 20<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[#February 20| Completions of Noncatenary Local Domains and UFDs]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 27<br />
| bgcolor="#C6D46E"| Sun Woo Park<br />
| bgcolor="#BCE2FE"|[[#February 27| Baker’s Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 6<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[#March 6| Mason's Conjectures and Chow Rings of Matroids]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 13<br />
| bgcolor="#C6D46E"| Brandon Boggess<br />
| bgcolor="#BCE2FE"|[[#March 13| Dial M_1,1 for moduli]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 27<br />
| bgcolor="#C6D46E"| Solly Parenti<br />
| bgcolor="#BCE2FE"|[[#March 27| Quadratic Forms]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 3<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[#April 3| Riemann-Roch and Abel-Jacobi theory on a finite graph]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 10<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[#April 10| Kindergarten GAGA]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 17<br />
| bgcolor="#C6D46E"| Soumya Sankar<br />
| bgcolor="#BCE2FE"|[[#April 17| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 24<br />
| bgcolor="#C6D46E"| Wendy Cheng<br />
| bgcolor="#BCE2FE"|[[#April 24| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| May 1<br />
| bgcolor="#C6D46E"| Shengyuan Huang<br />
| bgcolor="#BCE2FE"|[[#May 1| Orbifold Singular Cohomology]]<br />
|}<br />
</center><br />
<br />
== February 6 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vladimir Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Heisenberg Groups and the Fourier Transform<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will discuss the relationship between Fourier transforms and the Heisenberg groups, with a view toward the discussion of line bundles on complex tori that appears in Polishchuk's book Abelian Varieties, Theta functions, and the Fourier transform.<br />
<br />
|} <br />
</center><br />
<br />
== February 13 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''David Wagner'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: DG potpourri<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will explain how differential graded categories made their way into AG as a way to solve some of the inadequacies of the ordinary derived category. We will then give examples of the utility of DG techniques. <br />
<br /><br />
[[File:Dg-meme.png|center]]<br />
|} <br />
</center><br />
<br />
== February 20 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Completions of Noncatenary Local Domains and UFDs<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A ring is called catenary if for any pair of prime ideals P contained in Q, all saturated chains of prime ideals between P and Q have the same length.<br />
In this talk, I will introduce the necessary background about noncatenary rings and completions of local (Noetherian) domains, as well as the relevant history. Then, I will give the characterization of completions of noncatenary local domains and noncatenary local UFDs, which I will use to describe examples of very strange rings.<br />
|} <br />
</center><br />
<br />
== February 27 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Sun Woo Park'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Baker's Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: As a follow up talk to last semester, I will state and prove Baker’s theorem, a classical theorem which uses Newton Polygons to count the genus of a given curve. I will also briefly sketch how one can use Newton polygons to understand resolution of singularities of a given curve over valuation rings.<br />
<br /><br />
[[File:Sun_woo_baker.png|500px|center]]<br />
|} <br />
</center><br />
<br />
== March 6 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Mason's Conjectures and Chow Rings of Matroids<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A matroid is a combinatorial structure that abstracts many different notions of independence, including linear independence in a vector space. Mason's Conjectures are a series of three increasingly strong inequalities on certain numerical invariants of matroids, the weakest of which resisted proof for over 40 years until its resolution in 2015. However, in the years since, all of Mason's conjectures have become theorems!<br />
<br />
In this talk, we will introduce matroids and the Chow ring of a matroid, the amazing algebraic gadget used to prove Mason's first conjecture. Finally, outline the proof of Mason's second conjecture (work of our very own Botong Wang & coauthors).<br />
|} <br />
</center><br />
<br />
== March 13 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Brandon Boggess'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Dial M_1,1 for moduli<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: We will speculate wildly about what kind of object the moduli space of elliptic curves should be. I don't know what a stack is, and I promise not to try to define one.<br />
<br/><br />
[[File:Dial-M-For-Elliptic.png|400px|center]]<br />
|} <br />
</center><br />
<br />
== March 27 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Solly Parenti'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Quadratic Forms<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Starting from the classical story of binary quadratic forms, we'll move on to more modern aspects of the theory of quadratic forms and try to make sense of some mass formulas.<br />
|} <br />
</center><br />
<br />
== April 3 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Riemann-Roch and Abel-Jacobi theory on a finite graph<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A finite graph is like a Riemann surface, or a smooth projective curve. Following the paper by Baker and Norine, we will investigate linear equivalence of divisors on graphs, the Jacobian of a graph, and a combinatorial interpretation of these as a chip firing game.<br />
|} <br />
</center><br />
<br />
== April 10 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Kindergarten GAGA<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Join me in regressing to an infantile state as we discuss Serre's 1956 paper ''Algebraic geometry and analytic geometry'', widely considered to be the most influential work ever authored by a mathematician under the age of five. We will define the notion of an analytic space, construct the analytic space associated to any algebraic variety over <math>\mathbb{C}</math>, and examine the relationships between the two, including the equivalence between coherent algebraic sheaves and coherent analytic sheaves in the projective case.<br />
<br />
[[File:Badromancehof.png|500px|center]]<br />
|} <br />
</center><br />
<br />
== April 17 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 24 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Name'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== May 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Shengyuan Huang'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Orbifold Singular Cohomology<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/wisc.edu/cbooms/ Caitlyn Booms]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=17247Graduate Algebraic Geometry Seminar2019-03-31T00:13:50Z<p>Crowley: </p>
<hr />
<div>'''<br />
'''When:''' Wednesdays 4:25pm<br />
<br />
'''Where:''' Van Vleck B317 (Spring 2019)<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
<br />
'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cbooms@wisc.edu Caitlyn] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
<br />
==Ed Dewey's Wish List Of Olde==<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Famous Theorems===<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
__NOTOC__<br />
<br />
== Spring 2019 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| February 6<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[#February 6| Heisenberg Groups and the Fourier Transform]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 13<br />
| bgcolor="#C6D46E"| David Wagner<br />
| bgcolor="#BCE2FE"|[[#February 13| DG potpourri]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 20<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[#February 20| Completions of Noncatenary Local Domains and UFDs]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 27<br />
| bgcolor="#C6D46E"| Sun Woo Park<br />
| bgcolor="#BCE2FE"|[[#February 27| Baker’s Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 6<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[#March 6| Mason's Conjectures and Chow Rings of Matroids]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 13<br />
| bgcolor="#C6D46E"| Brandon Boggess<br />
| bgcolor="#BCE2FE"|[[#March 13| Dial M_1,1 for moduli]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 27<br />
| bgcolor="#C6D46E"| Solly Parenti<br />
| bgcolor="#BCE2FE"|[[#March 27| Quadratic Forms]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 3<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[#April 3| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 10<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[#April 10| Kindergarten GAGA]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 17<br />
| bgcolor="#C6D46E"| Soumya Sankar<br />
| bgcolor="#BCE2FE"|[[#April 17| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 24<br />
| bgcolor="#C6D46E"| Wendy Cheng<br />
| bgcolor="#BCE2FE"|[[#April 24| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| May 1<br />
| bgcolor="#C6D46E"| Shengyuan Huang<br />
| bgcolor="#BCE2FE"|[[#May 1| Orbifold Singular Cohomology]]<br />
|}<br />
</center><br />
<br />
== February 6 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vladimir Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Heisenberg Groups and the Fourier Transform<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will discuss the relationship between Fourier transforms and the Heisenberg groups, with a view toward the discussion of line bundles on complex tori that appears in Polishchuk's book Abelian Varieties, Theta functions, and the Fourier transform.<br />
<br />
|} <br />
</center><br />
<br />
== February 13 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''David Wagner'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: DG potpourri<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will explain how differential graded categories made their way into AG as a way to solve some of the inadequacies of the ordinary derived category. We will then give examples of the utility of DG techniques. <br />
<br /><br />
[[File:Dg-meme.png|center]]<br />
|} <br />
</center><br />
<br />
== February 20 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Completions of Noncatenary Local Domains and UFDs<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A ring is called catenary if for any pair of prime ideals P contained in Q, all saturated chains of prime ideals between P and Q have the same length.<br />
In this talk, I will introduce the necessary background about noncatenary rings and completions of local (Noetherian) domains, as well as the relevant history. Then, I will give the characterization of completions of noncatenary local domains and noncatenary local UFDs, which I will use to describe examples of very strange rings.<br />
|} <br />
</center><br />
<br />
== February 27 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Sun Woo Park'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Baker's Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: As a follow up talk to last semester, I will state and prove Baker’s theorem, a classical theorem which uses Newton Polygons to count the genus of a given curve. I will also briefly sketch how one can use Newton polygons to understand resolution of singularities of a given curve over valuation rings.<br />
<br /><br />
[[File:Sun_woo_baker.png|500px|center]]<br />
|} <br />
</center><br />
<br />
== March 6 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Mason's Conjectures and Chow Rings of Matroids<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A matroid is a combinatorial structure that abstracts many different notions of independence, including linear independence in a vector space. Mason's Conjectures are a series of three increasingly strong inequalities on certain numerical invariants of matroids, the weakest of which resisted proof for over 40 years until its resolution in 2015. However, in the years since, all of Mason's conjectures have become theorems!<br />
<br />
In this talk, we will introduce matroids and the Chow ring of a matroid, the amazing algebraic gadget used to prove Mason's first conjecture. Finally, outline the proof of Mason's second conjecture (work of our very own Botong Wang & coauthors).<br />
|} <br />
</center><br />
<br />
== March 13 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Brandon Boggess'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Dial M_1,1 for moduli<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: We will speculate wildly about what kind of object the moduli space of elliptic curves should be. I don't know what a stack is, and I promise not to try to define one.<br />
<br/><br />
[[File:Dial-M-For-Elliptic.png|400px|center]]<br />
|} <br />
</center><br />
<br />
== March 27 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Solly Parenti'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Quadratic Forms<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Starting from the classical story of binary quadratic forms, we'll move on to more modern aspects of the theory of quadratic forms and try to make sense of some mass formulas.<br />
|} <br />
</center><br />
<br />
== April 3 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Riemann-Roch and Abel-Jacobi theory on a finite graph<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A finite graph is like a Riemann surface, or a smooth projective curve. Following the paper by Baker and Norine, we will investigate linear equivalence of divisors on graphs, the Jacobian of a graph, and a combinatorial interpretation of these as a chip firing game.<br />
|} <br />
</center><br />
<br />
== April 10 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Kindergarten GAGA<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Join me in regressing to an infantile state as we discuss Serre's 1956 paper ''Algebraic geometry and analytic geometry'', widely considered to be the most influential work ever authored by a mathematician under the age of five. We will define the notion of an analytic space, construct the analytic space associated to any algebraic variety over <math>\mathbb{C}</math>, and examine the relationships between the two, including the equivalence between coherent algebraic sheaves and coherent analytic sheaves in the projective case.<br />
<br />
[[File:Badromancehof.png|500px|center]]<br />
|} <br />
</center><br />
<br />
== April 17 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 24 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Name'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== May 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Shengyuan Huang'''<br />
|-<br />
| bgcolor="#BCD2EE" | Title: Orbifold Singular Cohomology<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/wisc.edu/cbooms/ Caitlyn Booms]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Matroids_seminar&diff=17187Matroids seminar2019-03-20T18:51:34Z<p>Crowley: </p>
<hr />
<div>The matroids seminar & reading group meets '''10:00--10:45 on Fridays in Van Vleck 901''' in order to discuss matroids from a variety of viewpoints.<br />
In particular, we aim to<br />
* survey open conjectures and recent work in the area<br />
* compute many interesting examples<br />
* discover concrete applications<br />
<br />
We are happy to have new leaders of the discussion, and the wide range of topics to which matroids are related mean that each week is a great chance for a new participant to drop in! If you would like to talk but need ideas, see the [[Matroids seminar/ideas]] page.<br />
<br />
To help develop an inclusive environment, a subset of the organizers will be available before the talk in the ninth floor lounge to informally discuss background material e.g., "What is a variety?", "What is a circuit?", "What is a greedy algorithm?" (this is especially for those coming from an outside area).<br />
<br />
'''Organizers''': Colin Crowley, Connor Simpson; Daniel Corey, Jose Israel Rodriguez<br />
<br />
{| cellpadding="8px" style="vertical-align:top; text-align:left; cellpadding:10px;"<br />
|3/29/2019<br />
|<br />
<div style="font-weight:bold;">Colin Crowley</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Binary matroids and Seymour's decomposition in coding theory</i></div><br />
<div class="mw-collapsible-content"><br />
We will begin by discussing the equivalence between a binary matroid and a binary linear code. And then following [https://ac.els-cdn.com/009589568990052X/1-s2.0-009589568990052X-main.pdf?_tid=a1d8598e-c5f2-4c07-8d7b-0843e88c416f&acdnat=1552505468_33915774714b2e407e4fd7001ba33100 this paper] and [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4544972 this one], we will describe the Maximum Likelihood decoding problem and then outline how Seymour's decomposition theorem for regular matroids led to a polynomial time algorithm on a subclass of binary linear codes.<br />
</div></div><br />
|-<br />
|3/15/2019<br />
|<br />
<div style="font-weight:bold;">[https://sites.google.com/site/dcorey2814/home Daniel Corey]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>The geometry of thin Schubert cells</i></div><br />
<div class="mw-collapsible-content"><br />
We will cover the distinction between the thin Schubert cell of a matroid and the realization space of a matroid, how to compute examples, Mnev universality, and time permitting, maps between thin Schubert cells. <br />
</div></div><br />
|-<br />
|3/8/2019<br />
|<br />
<div style="font-weight:bold;">Vladmir Sotirov</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>is sick</i></div><br />
<div class="mw-collapsible-content"><br />
Plague and pestilence!<br />
</div></div><br />
|-<br />
|3/1/2019<br />
|<br />
<div style="font-weight:bold;">[https://math.berkeley.edu/~ceur/ Chris Eur]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>The multivariate Tutte polynomial of a flag matroid</i></div><br />
<div class="mw-collapsible-content"><br />
Flag matroids are combinatorial objects whose relation to ordinary matroids are akin to that of flag varieties to Grassmannians. We define a multivariate Tutte polynomial of a flag matroid, and show that it is Lorentzian in the sense of [https://arxiv.org/abs/1902.03719 Branden-Huh '19]. As a consequence, we obtain a flag matroid generalization of Mason’s conjecture concerning the f-vector of independent subsets of a matroid. This is an on-going joint work with June Huh.<br />
</div></div><br />
|-<br />
|2/22/2019<br />
|<br />
<div style="font-weight:bold;">[https://www.math.wisc.edu/~wang/ Botong Wang]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>The Kazhdan-Lusztig polynomial of a matroid</i></div><br />
<div class="mw-collapsible-content"><br />
Classically, Kazdhan-Lusztig polynomials are associated to intervals of the Bruhat poset of a Coxeter group. We will discuss an analogue of Kazdhan-Lusztig polynomials for matroids, including results and conjectures from [https://arxiv.org/pdf/1611.07474.pdf these] [https://arxiv.org/pdf/1412.7408.pdf two] papers. <br />
</div></div><br />
|-<br />
|2/15/2019<br />
|<br />
<div style="font-weight:bold;">Colin Crowley</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Matroid polytopes</i></div><br />
<div class="mw-collapsible-content"><br />
We outline the original formulation of matroid polytopes as moment polytopes of subvarieties of the Grassmanian, following [http://www.math.ias.edu/~goresky/pdf/combinatorial.jour.pdf Combinatorial Geometries, Convex Polyhedra, and Schbert Cells].<br />
</div></div><br />
|-<br />
|2/8/2019<br />
|<br />
<div style="font-weight:bold;">[http://www.math.wisc.edu/~csimpson6/ Connor Simpson]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Proving the Heron-Rota-Welsh conjecture</i></div><br />
<div class="mw-collapsible-content"><br />
We outline the proof of the Heron-Rota-Welsh conjecture given by Adiprasito, Huh, and Katz in their paper [https://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries]<br />
</div></div><br />
|-<br />
|1/25/2019 & 2/1/2019<br />
|<br />
<div style="font-weight:bold;">[https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Algebraic matroids in action</i></div><br />
<div class="mw-collapsible-content"><br />
We discuss algebraic matroids and their applications; see [https://arxiv.org/abs/1809.00865 Algebraic Matroids in Action].<br />
</div></div><br />
|-<br />
|1/18/2019<br />
|<br />
<div style="font-weight:bold;">[https://sites.google.com/site/dcorey2814/home Daniel Corey]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Introduction to matroids</i></div><br />
<div class="mw-collapsible-content"><br />
We'll cover the basic definitions and some examples, roughly following [http://web.ma.utexas.edu/users/sampayne/pdf/Math648Lecture3.pdf these notes].<br />
</div></div><br />
|-<br />
|}</div>Crowleyhttps://wiki.math.wisc.edu/index.php?title=Matroids_seminar&diff=17151Matroids seminar2019-03-13T20:16:24Z<p>Crowley: </p>
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<div>The matroids seminar & reading group meets '''10:00--10:45 on Fridays in Van Vleck 901''' in order to discuss matroids from a variety of viewpoints.<br />
In particular, we aim to<br />
* survey open conjectures and recent work in the area<br />
* compute many interesting examples<br />
* discover concrete applications<br />
<br />
We are happy to have new leaders of the discussion, and the wide range of topics to which matroids are related mean that each week is a great chance for a new participant to drop in! If you would like to talk but need ideas, see the [[Matroids seminar/ideas]] page.<br />
<br />
To help develop an inclusive environment, a subset of the organizers will be available before the talk in the ninth floor lounge to informally discuss background material e.g., "What is a variety?", "What is a circuit?", "What is a greedy algorithm?" (this is especially for those coming from an outside area).<br />
<br />
'''Organizers''': Colin Crowley, Connor Simpson; Daniel Corey, Jose Israel Rodriguez<br />
<br />
{| cellpadding="8px" style="vertical-align:top; text-align:left; cellpadding:10px;"<br />
|3/29/2019<br />
|<br />
<div style="font-weight:bold;">Colin Crowley</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Binary matroids and Seymour's decomposition in coding theory</i></div><br />
<div class="mw-collapsible-content"><br />
We will begin by discussing the equivalence between a binary matroid and a binary linear code. And then following [https://ac.els-cdn.com/009589568990052X/1-s2.0-009589568990052X-main.pdf?_tid=a1d8598e-c5f2-4c07-8d7b-0843e88c416f&acdnat=1552505468_33915774714b2e407e4fd7001ba33100 this paper] and [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4544972 this one], we will describe the Maximum Likelihood decoding problem and then outline how Seymour's decomposition theorem for binary matroids led to a polynomial time algorithm on a subclass of binary linear codes.<br />
</div></div><br />
|-<br />
|3/15/2019<br />
|<br />
<div style="font-weight:bold;">Daniel Corey</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Thin Schubert cells</i></div><br />
<div class="mw-collapsible-content"><br />
Tentatively, we will discuss thin schubert cells, their geometry, and maps between them in relation to faces of matroid polytopes. We will also give a statement of Mnev universality.<br />
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|-<br />
|3/8/2019<br />
|<br />
<div style="font-weight:bold;">Vladmir Sotirov</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>is sick</i></div><br />
<div class="mw-collapsible-content"><br />
Plague and pestilence!<br />
</div></div><br />
|-<br />
|3/1/2019<br />
|<br />
<div style="font-weight:bold;">[https://math.berkeley.edu/~ceur/ Chris Eur]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>The multivariate Tutte polynomial of a flag matroid</i></div><br />
<div class="mw-collapsible-content"><br />
Flag matroids are combinatorial objects whose relation to ordinary matroids are akin to that of flag varieties to Grassmannians. We define a multivariate Tutte polynomial of a flag matroid, and show that it is Lorentzian in the sense of [https://arxiv.org/abs/1902.03719 Branden-Huh '19]. As a consequence, we obtain a flag matroid generalization of Mason’s conjecture concerning the f-vector of independent subsets of a matroid. This is an on-going joint work with June Huh.<br />
</div></div><br />
|-<br />
|2/22/2019<br />
|<br />
<div style="font-weight:bold;">[https://www.math.wisc.edu/~wang/ Botong Wang]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>The Kazhdan-Lusztig polynomial of a matroid</i></div><br />
<div class="mw-collapsible-content"><br />
Classically, Kazdhan-Lusztig polynomials are associated to intervals of the Bruhat poset of a Coxeter group. We will discuss an analogue of Kazdhan-Lusztig polynomials for matroids, including results and conjectures from [https://arxiv.org/pdf/1611.07474.pdf these] [https://arxiv.org/pdf/1412.7408.pdf two] papers. <br />
</div></div><br />
|-<br />
|2/15/2019<br />
|<br />
<div style="font-weight:bold;">Colin Crowley</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Matroid polytopes</i></div><br />
<div class="mw-collapsible-content"><br />
We outline the original formulation of matroid polytopes as moment polytopes of subvarieties of the Grassmanian, following [http://www.math.ias.edu/~goresky/pdf/combinatorial.jour.pdf Combinatorial Geometries, Convex Polyhedra, and Schbert Cells].<br />
</div></div><br />
|-<br />
|2/8/2019<br />
|<br />
<div style="font-weight:bold;">[http://www.math.wisc.edu/~csimpson6/ Connor Simpson]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Proving the Heron-Rota-Welsh conjecture</i></div><br />
<div class="mw-collapsible-content"><br />
We outline the proof of the Heron-Rota-Welsh conjecture given by Adiprasito, Huh, and Katz in their paper [https://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries]<br />
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|-<br />
|1/25/2019 & 2/1/2019<br />
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<div style="font-weight:bold;">[https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Algebraic matroids in action</i></div><br />
<div class="mw-collapsible-content"><br />
We discuss algebraic matroids and their applications; see [https://arxiv.org/abs/1809.00865 Algebraic Matroids in Action].<br />
</div></div><br />
|-<br />
|1/18/2019<br />
|<br />
<div style="font-weight:bold;">[https://sites.google.com/site/dcorey2814/home Daniel Corey]</div><br />
<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"><br />
<div><i>Introduction to matroids</i></div><br />
<div class="mw-collapsible-content"><br />
We'll cover the basic definitions and some examples, roughly following [http://web.ma.utexas.edu/users/sampayne/pdf/Math648Lecture3.pdf these notes].<br />
</div></div><br />
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|}</div>Crowley