Graduate Algebraic Geometry Seminar Spring 2023: Difference between revisions

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'''When:''' 1:30-2:30 PM on Fridays
'''When:''' 4:15-5:15 PM on Wednesday.


'''Where:''' Van Vleck B219
'''Where:''' Van Vleck B119
[[Image:newcat.jpg|thumb|220px| Toby the OFFICIAL mascot of GAGS!!]]
[[Image:newcat.jpg|thumb|220px| Toby the OFFICIAL mascot of GAGS!!]]


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'''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].
'''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].


''' Organizers: ''' [https://johndcobb.github.io John Cobb], Yu (Joey) Luo
''' Organizers: ''' [https://johndcobb.github.io John Cobb], [https://people.math.wisc.edu/~yluo237/ Yu (Joey) Luo]


==Give a talk!==
==Give a talk!==
We need volunteers to give talks this semester. 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]. Sign up here: https://forms.gle/XUAq1VFFqqErKDEh6.
We need volunteers to give talks this semester. 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]. Sign up [https://forms.gle/XmdJ4hGbxvpbVVwr5 here].


===Fall 2022 Topic Wish List===
===Wishlist===
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.
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.
* Hilbert Schemes
* Hilbert Schemes
*Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
*Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
*Going from line bundles and divisors to vector bundles and chern classes
*A History of the Weil Conjectures
*A History of the Weil Conjectures
*Mumford & Bayer, "What can be computed in Algebraic Geometry?"
*A pre talk for any other upcoming talk
*A pre talk for any other upcoming talk
*Weil Conjectures, GAGA theorems, surfaces of general type, moduli spaces, moduli of curves, mixed characteristics (stuff), elliptic curves, abelian varieties, hyperelliptic curves, resolution of singularities, minimal model program (stuff).
*Weil Conjectures, GAGA theorems, surfaces of general type, moduli spaces, moduli of curves, mixed characteristics (stuff), elliptic curves, abelian varieties, hyperelliptic curves, resolution of singularities, minimal model program (stuff).
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| bgcolor="#BCD2EE" width="300" align="center" |'''Title'''
| bgcolor="#BCD2EE" width="300" align="center" |'''Title'''
|-
|-
| bgcolor="#E0E0E0" |September 23
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#January 31|January 31]]
| bgcolor="#C6D46E" | Mahrud Sayrafi
| bgcolor="#BCE2FE" | Bounding the Multigraded Regularity of Powers of Ideals
|-
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#February 1|February 1]]
| bgcolor="#C6D46E" |John Cobb
| bgcolor="#BCE2FE" |Introduction to Intersection Theory
|-
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#February 8|February 8]]
| bgcolor="#C6D46E" |Yiyu Wang
| bgcolor="#C6D46E" |Yiyu Wang
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#September 23|The Cox Ring of Toric Varieties]]
| bgcolor="#BCE2FE" |An introduction to Macpherson's Chern classes
|-
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#February 15|February 15]]
| bgcolor="#C6D46E" |Alex Hof
| bgcolor="#BCE2FE" |Normal Cones in Algebraic Geometry
|-
|-
| bgcolor="#E0E0E0" |September 30
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#February 22|February 22]]
| bgcolor="#C6D46E" |Asvin G.
| bgcolor="#C6D46E" |Maya Banks
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#September 30|The moduli space of curves]]
| bgcolor="#BCE2FE" |Syzygies of Projective Varieties
|-
|-
| bgcolor="#E0E0E0" |October 7
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#March 3|March 1]]
| bgcolor="#C6D46E" |Alex Hof
| bgcolor="#C6D46E" |Asvin G
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#October 7|Revenge of the Classical Topology]]
| bgcolor="#BCE2FE" |TBD
|-
|-
| bgcolor="#E0E0E0" |October 14
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#March 10|March 8]]
| bgcolor="#C6D46E" |John Cobb
| bgcolor="#C6D46E" |
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#October 14|Virtual Resolutions and Syzygies]]
| bgcolor="#BCE2FE" |
|-
|-
| bgcolor="#E0E0E0" |October 21
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#March 24|March 22]]
| bgcolor="#C6D46E" |Yifan
| bgcolor="#C6D46E" |Kevin Dao
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#October 21|TBA]]
| bgcolor="#BCE2FE" | Enriques-Kodaira Classification and its Influence on MMP
|-
|-
| bgcolor="#E0E0E0" |October 28
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#March 31|March 29]]
| bgcolor="#C6D46E" |Ivan Aidun
| bgcolor="#C6D46E" |Peter Yi Wei
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#October 28|Gröbner Bases and Computations in Algebraic Geometry]]
| bgcolor="#BCE2FE" |TBD
|-
|-
| bgcolor="#E0E0E0" |November 2
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#April 7|April 5]]
| bgcolor="#C6D46E" |Some Matroid Person
| bgcolor="#C6D46E" |Dima Arinkin
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#November 2|TBA]]
| bgcolor="#BCE2FE" |Hitchin Fibration
|-
|-
| bgcolor="#E0E0E0" | November 11
| bgcolor="#E0E0E0" |[[Graduate Algebraic Geometry Seminar Spring 2023#April 14|April 12]]
| bgcolor="#C6D46E" |Connor Simpson
| bgcolor="#C6D46E" |Yunfan He
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#November 11|A miraculous theorem of Brion]]
| bgcolor="#BCE2FE" |Variation of Hodge structure
|-
|-
| bgcolor="#E0E0E0" |November 18
| bgcolor="#E0E0E0" | [[Graduate Algebraic Geometry Seminar Spring 2023#April 21|April 19]]
| bgcolor="#C6D46E" |Alex Mine
| bgcolor="#C6D46E" | Jacob Wood
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#November 18|Singular curves]]
| bgcolor="#BCE2FE" |K-Theory or something
|-
|-
| bgcolor="#E0E0E0" |December 2
| bgcolor="#E0E0E0" | [[Graduate Algebraic Geometry Seminar Spring 2023#April 28|April 26]]
| bgcolor="#C6D46E" |Kevin Dao
| bgcolor="#C6D46E" | Brian Hepler
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#December 2|Commentary on Local Cohomology]]
| bgcolor="#BCE2FE" |Condensed Sets
|-
|-
| bgcolor="#E0E0E0" | December 9
| bgcolor="#E0E0E0" | [[Graduate Algebraic Geometry Seminar Spring 2023#May 5|May 3]]
| bgcolor="#C6D46E" | Yu LUO
| bgcolor="#C6D46E" | Sun Woo Park
| bgcolor="#BCE2FE" |[[Graduate Algebraic Geometry Seminar Fall 2022#December 9|Yoneda Embedding and Moduli Problems]]
| bgcolor="#BCE2FE" |Introduction to Newton Polygon
|}
|}
</center>
</center>


===September 23===
===January 31===
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | Mahrud Sayrafi
|-
| bgcolor="#BCD2EE" align="center" |Title: Bounding the Multigraded Regularity of Powers of Ideals
|-
| bgcolor="#BCD2EE" |Abstract: Building on a result of Swanson, Cutkosky-Herzog-Trung and Kodiyalam described the surprisingly predictable asymptotic behavior of Castelnuovo--Mumford regularity for powers of ideals on a projective space P^n: given an ideal I, there exist integers d and e such that for large enough n the regularity of I^n is exactly dn+e.
Through a medley of examples we will see why asking the same question about an ideal I in the total coordinate ring S of a smooth projective toric variety X is interesting. After that I will summarize the ideas and methods we used to bound the region reg(I^n) as a subset of Pic(X) by proving that it contains a translate of reg(S) and is contained in a translate of Nef(X), with each bound translating by a fixed vector as n increases. Along the way will see some surprising behavior for multigraded regularity of modules. This is joint work with Juliette Bruce and Lauren Cranton Heller.
|}                                                                       
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===February 1===
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{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | John Cobb
|-
| bgcolor="#BCD2EE" align="center" |Title: Introduction to Intersection Theory
|-
| bgcolor="#BCD2EE" |Abstract: In this <s>advertisement</s> talk, I'd like to talk about some methods used in enumerative geometry. I'll define what a Chow ring is, count some things with it, and tell you why you should read "3264 and all that" this semester with me.
|}                                                                       
</center>
 
===February 8===
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<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
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| bgcolor="#A6B658" align="center" style="font-size:125%" | Yiyu Wang
| bgcolor="#A6B658" align="center" style="font-size:125%" | Yiyu Wang
|-
|-
| bgcolor="#BCD2EE" align="center" |Title: The Cox Ring of Toric Varieties
| bgcolor="#BCD2EE" align="center" |Title: An introduction to Macpherson's Chern classes
|-
|-
| bgcolor="#BCD2EE" |Abstract: This talk will include two parts. In the first part, I will briefly introduce toric varieties, and give some examples. I will also explain how they are related to the combinatorial objects called fans. Only some basic algebraic geometry will be used in this part. In the second part, I will talk about Cox's construction of representing any toric variety as a quotient space, and his famous Cox ring. As a corollary, we can prove that the automorphism group of a complete simplicial toric variety is a linear algebraic group. I will use some basic knowledge of toric varieties in this part.
| bgcolor="#BCD2EE" |Abstract: In this talk, I will start from a formula of the Euler characteristic number of a degree d smooth hypersurface in P^n and discuss how to generalize this formula to the singular case. This naturally leads to the notion of the Chern classes of a singular space. I will briefly introduce Macpherson's Chern classes which is a natural generalization of the ordinary Chern class and how to calculate these classes.
|}                                                                         
|}                                                                         
</center>
</center>


===September 30===
===February 15===
<center>
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |Asvin G
| bgcolor="#A6B658" align="center" style="font-size:125%" |Alex Hof
|-
| bgcolor="#BCD2EE" align="center" |Title: Normal Cones in Algebraic Geometry
|-
| bgcolor="#BCD2EE" |Abstract: In this talk, we'll go over the definition of the normal cone of a closed subscheme, explore the geometric intuition behind it via a construction called the Rees algebra, and explain how it can be used to give geometric characterizations of apparently algebraic notions such as flatness and depth.
|}                                                                       
</center><center></center>
 
===February 22===
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |Maya Banks
|-
|-
| bgcolor="#BCD2EE" align="center" |Title: The moduli space of curves
| bgcolor="#BCD2EE" align="center" |Title: Syzygies of Projective Varieties
|-
|-
| bgcolor="#BCD2EE" |Abstract: I'll give a brief introduction to moduli spaces and focus mostly on the moduli spaces of genus 0 curves with marked points. These spaces are at the same time quite explicit and easy to describe while also having connections with many interesting parts of mathematics. I'll try to keep the topic fairly elementary.
| bgcolor="#BCD2EE" | Abstract: The general slogan for the study of syzygies in geometry is that "geometric information about a projective variety is reflected in its sygygies." In this talk, we'll discuss some of the early results that kick-started this idea, such as Castelnuovo-Mumford regularity, quadric generation of varieties in P^n, and Green's Linear Syzygy Theorem. I'll go over all of the basic definitions and hopefully do lots of examples---in particular, this talk should be accessible to someone taking the intro Algebraic Geometry sequence.
|}                                                                         
|}                                                                         
</center>
</center>


===October 7===
===March 1===
<center>
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |Alex Hof
| bgcolor="#A6B658" align="center" style="font-size:125%" |Asvin G
|-
|-
| bgcolor="#BCD2EE" align="center" |Title: Revenge of the Classical Topology
| bgcolor="#BCD2EE" align="center" |Title: TBD
|-
|-
| bgcolor="#BCD2EE" |Abstract: The Zariski topology is pretty cool, but, if we're working over the complex numbers, we can also think about the classical topology we're used to from other areas of math. In this talk, we'll discuss analytification, the process of passing from an algebraic variety (or scheme) to the corresponding classical object (or complex-analytic space), and touch on various facts about the relationship between the two, such as Serre's GAGA principle. If time permits, we'll also talk a little about tools we can use to gain insight about varieties once we've analytified them, such as the theory of stratifications.
| bgcolor="#BCD2EE" |Abstract:
|}                                                                         
|}                                                                         
</center>
</center>


=== October 14===
===March 8===
<center>
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |John Cobb
| bgcolor="#A6B658" align="center" style="font-size:125%" |
|-
|-
| bgcolor="#BCD2EE" align="center" |Title: Virtual Resolutions and Syzygies
| bgcolor="#BCD2EE" align="center" |Title:  
|-
|-
| bgcolor="#BCD2EE" | Abstract: Starting from polynomials and proceeding with specific examples, the first part of this talk is dedicated to motivating the idea of syzygies and the geometric information they encode about projective varieties. This will lead us to a current focus of research: How can we use these important tools when our variety is not projective? At least in the situation of toric varieties, we can use a generalization called ''virtual'' syzygies. The second part of the talk will focus on answering a few basic questions about these analogues, such as: How can we construct examples? How complicated do they get (at least in the case of curves)?
| bgcolor="#BCD2EE" |Abstract:  
|}                                                                         
|}                                                                         
</center>
</center>


===October 21===
===March 22===
<center>
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |Yifan
| bgcolor="#A6B658" align="center" style="font-size:125%" |Kevin Dao
|-
|-
| bgcolor="#BCD2EE" align="center" |Title:
| bgcolor="#BCD2EE" align="center" |Title: Enriques-Kodaira Classification and its Influence on MMP
|-
|-
| bgcolor="#BCD2EE" |Abstract:
| bgcolor="#BCD2EE" |Abstract: There is always more to say than there is time to say it. Let this abstract be an overly optimistic summary. I’ll tell you what the EK classification is, how it is achieved, the relevant development of birational algebraic geometry, and then point towards  the difficulties in higher dimensions. I’ll also indicate, where possible and from what I know, the technical tools that are ubiquitous to the topic. If there is time, I will indicate a few problems and directions in either (a) the classification of (non-algebraic) surfaces, (b) rational curves on varieties, (c) major results of the MMP itself.
|}                                                                         
|}                                                                         
</center>
</center>


=== October 28===
===March 29===
<center>
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |Ivan Aidun
| bgcolor="#A6B658" align="center" style="font-size:125%" |Peter Yi Wei
|-
|-
| bgcolor="#BCD2EE" align="center" |Title: Gröbner Bases and Computations in Algebraic Geometry
| bgcolor="#BCD2EE" align="center" |Title: TBD
|-
|-
| bgcolor="#BCD2EE" |Abstract: Gröbner bases are the most important computational tools in algebraic geometry and commutative algebra.  They can be computed by an algorithm which simultaneously generalizes row reduction for matrices and the Eucliden algorithm for polynomial division.  This algorithm has two peculiar properties: its worst-case time complexity is doubly exponential in the number of variables, but it also runs quickly on most examples of practical interest.  What could account for the difference between the expected and actual runtime of this algorithm?
| bgcolor="#BCD2EE" |Abstract:  
|}                                                                         
|}                                                                         
</center>
</center>


===November 2===
===April 5===
<center>
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |Some Matroid Person
| bgcolor="#A6B658" align="center" style="font-size:125%" |Dima Arinkin
|-
|-
| bgcolor="#BCD2EE" align="center" |Title:
| bgcolor="#BCD2EE" align="center" |Title: Hitchin Fibration
 
|-
|-
| bgcolor="#BCD2EE" |Abstract:
| bgcolor="#BCD2EE" |Abstract:
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</center>
</center>


===November 11===
===April 12===
<center>
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |Connor Simpson
| bgcolor="#A6B658" align="center" style="font-size:125%" |Yunfan He
|-
|-
| bgcolor="#BCD2EE" align="center" |Title: A miraculous theorem of Brion
| bgcolor="#BCD2EE" align="center" |Title: Variation of Hodge structure
|-
|-
| bgcolor="#BCD2EE" |Abstract: Schubert varieties give a basis for the cohomology ring of the grassmannian. We'll discuss a heorem of Brion, which says that subvarieties of the grassmannian whose expression in the schubert basis uses only 0 and 1 coefficients have numerous nice properties.
| bgcolor="#BCD2EE" |Abstract: I will review a little bit of the classic Hodge theory, and talk about how to generalize to the relative version.
|}                                                                         
|}                                                                         
</center>
</center>


===November 18===
===April 19===
<center>
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |Alex Mine
| bgcolor="#A6B658" align="center" style="font-size:125%" |Jacob Wood
|-
|-
| bgcolor="#BCD2EE" align="center" |Title: Singular curves
| bgcolor="#BCD2EE" align="center" |Title:
 
|-
|-
| bgcolor="#BCD2EE" |Abstract: I’ll say a few things that I know about singular curves and their compactified Jacobians, and maybe even a few things that I don’t.
| bgcolor="#BCD2EE" |Abstract:  
|}                                                                         
|}                                                                         
</center>
</center>


===December 2===
===April 26===
<center>
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |Kevin Dao
| bgcolor="#A6B658" align="center" style="font-size:125%" |Brian Hepler
|-
|-
| bgcolor="#BCD2EE" align="center" |Title: Commentary on Local Cohomology
| bgcolor="#BCD2EE" align="center" |Title:
|-
|-
| bgcolor="#BCD2EE" |Abstract: Local cohomology is sheaf cohomology with supports which provides local information. Applications include the proof of Hard Lefschetz Theorems, bounds on the generators of a radical ideals, and results regarding connectedness of varieties like the Fulton-Hansen Theorem. For this talk, I will define the local cohomology functors, describe different ways to define it, properties of local cohomology, the punctured spectrum, and some applications. If time permits, will sketch the proof of the Fulton-Hansen Theorem.
| bgcolor="#BCD2EE" |Abstract:  
|}                                                                         
|}                                                                         
</center>
</center>


===December 9===
===May 3===
<center>
<center>
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
{| style="color:black; font-size:100%" border="2" cellpadding="10" width="700" cellspacing="20" table
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" |Yu (Joey) LUO
| bgcolor="#A6B658" align="center" style="font-size:125%" |Sun Woo Park
|-
|-
| bgcolor="#BCD2EE" align="center" |Title: Yoneda Embedding and Moduli Problems
| bgcolor="#BCD2EE" align="center" |Title: Introduction to Newton Polygon
|-
|-
| bgcolor="#BCD2EE" |Abstract: This will be a talk intended for the first year algebraic geometers. I'll start with Yoneda embedding and "functor of points", and briefly discuss what we can do under this setting. Then I will talk about the moduli problems. For each part I will give some examples, which example I will give is depends on how many scheme theory I'm allowed to use.
| bgcolor="#BCD2EE" |Abstract:  
If time permits, I can talk about some generalization, depends on how many fancy language I'm allowed to use.
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==Past Semesters==
==Past Semesters==
[https://wiki.math.wisc.edu/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2022 Fall 2022]
[https://wiki.math.wisc.edu/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2022 Spring 2022]
[https://wiki.math.wisc.edu/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2022 Spring 2022]


Latest revision as of 21:09, 12 April 2023

When: 4:15-5:15 PM on Wednesday.

Where: Van Vleck B119

Toby the OFFICIAL mascot of GAGS!!

Who: All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.

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 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.

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 here.

Organizers: John Cobb, Yu (Joey) Luo

Give a talk!

We need volunteers to give talks this semester. 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 main page. Sign up here.

Wishlist

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.

  • Hilbert Schemes
  • Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
  • A History of the Weil Conjectures
  • A pre talk for any other upcoming talk
  • Weil Conjectures, GAGA theorems, surfaces of general type, moduli spaces, moduli of curves, mixed characteristics (stuff), elliptic curves, abelian varieties, hyperelliptic curves, resolution of singularities, minimal model program (stuff).

Being an audience member

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:

  • Do Not Speak For/Over the Speaker
  • Ask Questions Appropriately

Talks

Date Speaker Title
January 31 Mahrud Sayrafi Bounding the Multigraded Regularity of Powers of Ideals
February 1 John Cobb Introduction to Intersection Theory
February 8 Yiyu Wang An introduction to Macpherson's Chern classes
February 15 Alex Hof Normal Cones in Algebraic Geometry
February 22 Maya Banks Syzygies of Projective Varieties
March 1 Asvin G TBD
March 8
March 22 Kevin Dao Enriques-Kodaira Classification and its Influence on MMP
March 29 Peter Yi Wei TBD
April 5 Dima Arinkin Hitchin Fibration
April 12 Yunfan He Variation of Hodge structure
April 19 Jacob Wood K-Theory or something
April 26 Brian Hepler Condensed Sets
May 3 Sun Woo Park Introduction to Newton Polygon

January 31

Mahrud Sayrafi
Title: Bounding the Multigraded Regularity of Powers of Ideals
Abstract: Building on a result of Swanson, Cutkosky-Herzog-Trung and Kodiyalam described the surprisingly predictable asymptotic behavior of Castelnuovo--Mumford regularity for powers of ideals on a projective space P^n: given an ideal I, there exist integers d and e such that for large enough n the regularity of I^n is exactly dn+e.

Through a medley of examples we will see why asking the same question about an ideal I in the total coordinate ring S of a smooth projective toric variety X is interesting. After that I will summarize the ideas and methods we used to bound the region reg(I^n) as a subset of Pic(X) by proving that it contains a translate of reg(S) and is contained in a translate of Nef(X), with each bound translating by a fixed vector as n increases. Along the way will see some surprising behavior for multigraded regularity of modules. This is joint work with Juliette Bruce and Lauren Cranton Heller.

February 1

John Cobb
Title: Introduction to Intersection Theory
Abstract: In this advertisement talk, I'd like to talk about some methods used in enumerative geometry. I'll define what a Chow ring is, count some things with it, and tell you why you should read "3264 and all that" this semester with me.

February 8

Yiyu Wang
Title: An introduction to Macpherson's Chern classes
Abstract: In this talk, I will start from a formula of the Euler characteristic number of a degree d smooth hypersurface in P^n and discuss how to generalize this formula to the singular case. This naturally leads to the notion of the Chern classes of a singular space. I will briefly introduce Macpherson's Chern classes which is a natural generalization of the ordinary Chern class and how to calculate these classes.

February 15

Alex Hof
Title: Normal Cones in Algebraic Geometry
Abstract: In this talk, we'll go over the definition of the normal cone of a closed subscheme, explore the geometric intuition behind it via a construction called the Rees algebra, and explain how it can be used to give geometric characterizations of apparently algebraic notions such as flatness and depth.

February 22

Maya Banks
Title: Syzygies of Projective Varieties
Abstract: The general slogan for the study of syzygies in geometry is that "geometric information about a projective variety is reflected in its sygygies." In this talk, we'll discuss some of the early results that kick-started this idea, such as Castelnuovo-Mumford regularity, quadric generation of varieties in P^n, and Green's Linear Syzygy Theorem. I'll go over all of the basic definitions and hopefully do lots of examples---in particular, this talk should be accessible to someone taking the intro Algebraic Geometry sequence.

March 1

Asvin G
Title: TBD
Abstract:

March 8

Title:
Abstract:

March 22

Kevin Dao
Title: Enriques-Kodaira Classification and its Influence on MMP
Abstract: There is always more to say than there is time to say it. Let this abstract be an overly optimistic summary. I’ll tell you what the EK classification is, how it is achieved, the relevant development of birational algebraic geometry, and then point towards the difficulties in higher dimensions. I’ll also indicate, where possible and from what I know, the technical tools that are ubiquitous to the topic. If there is time, I will indicate a few problems and directions in either (a) the classification of (non-algebraic) surfaces, (b) rational curves on varieties, (c) major results of the MMP itself.

March 29

Peter Yi Wei
Title: TBD
Abstract:

April 5

Dima Arinkin
Title: Hitchin Fibration
Abstract:

April 12

Yunfan He
Title: Variation of Hodge structure
Abstract: I will review a little bit of the classic Hodge theory, and talk about how to generalize to the relative version.

April 19

Jacob Wood
Title:
Abstract:

April 26

Brian Hepler
Title:
Abstract:

May 3

Sun Woo Park
Title: Introduction to Newton Polygon
Abstract:

Past Semesters

Fall 2022

Spring 2022

Fall 2021

Spring 2021

Fall 2020

Spring 2020

Fall 2019

Spring 2019

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

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