https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Cgeske&feedformat=atomUW-Math Wiki - User contributions [en]2022-09-29T11:50:36ZUser contributionsMediaWiki 1.35.6https://wiki.math.wisc.edu/index.php?title=Madison_Math_Circle&diff=15961Madison Math Circle2018-09-11T21:01:19Z<p>Cgeske: /* Meetings for Fall 2018 */</p>
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<div>[[Image:logo.png|right|440px]]<br />
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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 />
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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 />
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[[Image: MathCircle_2.jpg|500px]] [[Image: MathCircle_4.jpg|500px]] <br />
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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 />
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=All right, I want to come!=<br />
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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 />
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[https://uwmadison.co1.qualtrics.com/jfe/form/SV_e9WdAs2SXNurWFD '''Math Circle Registration Form''']<br />
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All of your information is kept private, and is only used by the Madison Math Circle organizer to help run the Circle. <br />
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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 />
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==Directions and parking==<br />
Our meetings are held on the 3rd floor of Helen C. White Hall in room 3255.<br />
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<div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><br />
[[File: Helencwhitemap.png|400px]]</div><br />
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'''Parking.''' Parking on campus is rather limited. Here is as list of some options:<br />
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*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 />
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==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 />
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==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:mathcircleorganizers@lists.wisc.edu here]. We are always interested in feedback!<br />
<center><br />
<gallery widths=480px heights=240px 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]<br />
File:Ee.jpg|[http://www.math.wisc.edu/~evaelduque/ Eva Elduque]<br />
File:mrjulian.jpg|[http://www.math.wisc.edu/~mrjulian/ Ryan Julian]<br />
File:soumyasankar.jpg|[http://www.math.wisc.edu/~soumyasankar Soumya Sankar]<br />
</gallery><br />
</center><br />
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==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 />
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[http://www.math.wisc.edu/donate Online Donation Link]<br />
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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 />
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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 />
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Or you can just pay in cash, and we'll give you a receipt.<br />
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==Help us grow!==<br />
If you like Math Circle, please help us continue to grow! Students, parents, and teachers can help by:<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 2018=<br />
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<center><br />
<br />
Unless specified talks start at '''6pm in room 3255 of Helen C. White Library''', unless otherwise noted.<br />
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{| style="color:black; font-size:120%" border="1" cellpadding="14" cellspacing="0"<br />
|-<br />
! colspan="3" style="background: #e8b2b2;" align="center" | Fall 2018<br />
|-<br />
! Date !! Speaker !! Topic<br />
|-<br />
| September 17, 2018 || Sun Woo Park || TBD<br />
|-<br />
| September 24, 2018 || TBD || TBD<br />
|-<br />
| October 1, 2018 || TBD || TBD<br />
|-<br />
| October 8, 2018 || TBD || TBD<br />
|-<br />
| October 15, 2018 || TBD || TBD<br />
|-<br />
| October 22, 2018 || TBD || TBD<br />
|-<br />
| October 29, 2018 || TBD || TBD<br />
|-<br />
| November 5, 2018 || Christian Geske || Josephus Problem<br />
|-<br />
| November 12, 2018 || TBD || TBD<br />
|-<br />
| November 19, 2018 || TBD || TBD<br />
|-<br />
|}<br />
<br />
</center><br />
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=Off-Site Meetings=<br />
<br />
We will hold some Math Circle meetings at local high schools on early release days. Our schedule for 2018-2019has not yet been determined. If you are interesting in having us come to your high school, please contact us!<br />
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<!--<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 2017<br />
|-<br />
|-<br />
! Date !! Time !! Location !! Speaker !! Topic <br />
|-<br />
| October 2nd || 2:45pm|| East High School - Madison, WI || [http://www.math.wisc.edu/~moises/ Moisés Herradón] || [https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#October_2_2017_.28East_High_School.29 How to make it as a Hackenbush player in the planet Zubenelgenubi 4] <br />
|-<br />
| November 2nd || 9:30am || Wisconsin Institute for Discovery - Madison, WI || [http://www.math.wisc.edu/~rzachariah/ Alisha Zachariah] || <br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#November_2_2017_.28WID.29_-_1 Fractals, Fractions and Fibonnaci.]<br />
|-<br />
| November 2nd (Canceled) || 11:00am || Wisconsin Institute for Discovery - Madison, WI || [http://zachcharles.wordpress.com Zach Charles] || <br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#November_2_2017_.28WID.29_-_2 1+1 = 10 or "How does my computer do anything?"]<br />
|-<br />
| November 2nd || 6:00pm || Whitewater High School - Whitewater, WI || [http://www.math.wisc.edu/~juliettebruce Juliette Bruce] || <br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#November_2_2017_.28Whitewater.29 Doodling Daydreams]<br />
|-<br />
| November 3rd || 10:30am || KM Global - Wales, WI || [http://www.math.wisc.edu/~stovall/ Betsy Stovall] ||<br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#November_3_2017_.28KM_Global.29 Recent discoveries in mathematics]<br />
|-<br />
| November 27th || 2:45pm|| JMM High School - Madison, WI || [http://www.math.wisc.edu/~juliettebruce Juliette Bruce] || <br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#November_27_2017_.28JMM_High_School.29 Is any knot not the unknot?]<br />
|-<br />
| December 11th || 2:45pm|| East High School - Madison, WI || [https://sites.google.com/wisc.edu/jwg John Wiltshire-Gordon] ||<br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#December_11_2017_.28East_High_School.29 What if seven is zero?]<br />
|-<br />
<br />
! colspan="5" style="background: #e8b2b2;" align="center" | Spring 2018<br />
|-<br />
|-<br />
! Date !! Time !! Location !! Speaker !! Topic <br />
|-<br />
| February 19th || 2:45pm|| East High School - Madison, WI || Jordan Ellenberg || [https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#February_19_2018_.28East_High_School.29 Is math destroying the right to vote?] <br />
|-<br />
| March 1st || 9:30am || Wisconsin Institute for Discovery - Madison, WI || Wanlin Li || <br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#March_1_2018_.28WID.29_-_1 From Patterns to Functions?]<br />
|-<br />
| March 1st || 10:45am || Wisconsin Institute for Discovery - Madison, WI || Daniel Erman || <br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#March_1_2018_.28WID.29_-_2 What does math research look like?]<br />
|-<br />
| March 5th || 2:45pm|| JMM High School - Madison, WI || Zach Charles || [https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#March_5_2018_.28JMM_High_School.29 1+1 = 10 or "How does my computer do anything?"]<br />
|-<br />
| March 19th || 2:45pm|| East High School - Madison, WI || Phil Wood || [https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#March_19_2018_.28East_High_School.29 The Mathematics of Winning Strategies ] <br />
|-<br />
| April 30th || 2:45pm|| JMM High School - Madison, WI || Juliette Bruce || [https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts#April_30_2018_.28JMM_High_School.29 From Books to Mars]<br />
|-<br />
|}<br />
</center><br />
--><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 />
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[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts_2016-2017 2016 - 2017 Abstracts]<br />
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[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_2015-2016 2015 - 2016 Math Circle Page]<br />
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[https://www.math.wisc.edu/wiki/index.php/Math_Circle_de_Madison_2015-2016 2015 - 2016 Math Circle Page (Spanish)]<br />
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[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.mathcircles.org/content/lesson-plans Sample Lesson Plans]<br />
<br />
[http://www.mathcircles.org/content/circle-box "Circle in a Box"]</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=14505Graduate/Postdoc Topology and Singularities Seminar2017-11-02T21:41:51Z<p>Cgeske: /* Fall 2017 */</p>
<hr />
<div>== Fall 2017==<br />
<br />
The Seminar meets at 3:30 to 4:30 pm on Wednesdays in Van Vleck 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Oct 4<br />
|Eva Elduque <br />
|"Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (I)"<br />
|-<br />
|-<br />
|Oct 11<br />
|Eva Elduque <br />
|"Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (II)"<br />
|-<br />
|-<br />
|Oct 18<br />
|Sebastian Baader <br />
|"Dehn twist length in mapping class groups"<br />
|-<br />
|-<br />
|Oct 25<br />
|Cancelled <br />
|-<br />
|-<br />
|Nov 1<br />
|Christian Geske <br />
|"Algebraic Intersection Spaces (I)"<br />
|-<br />
|-<br />
|Nov 8<br />
|Christian Geske <br />
|"Algebraic Intersection Spaces (II)"<br />
|}<br />
<br />
== Spring 2017==<br />
Fridays at 11:00 VV901<br />
<br />
The Seminar meets on Fridays at 11:00 pm in Van Vleck 901, and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Jan 27<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation I"<br />
|-<br />
|Feb 3<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation II"<br />
|-<br />
|Feb 10<br />
|Sashka <br />
|"The Wirtinger Number of a knot equals its bridge number I"<br />
|-<br />
|Feb 17<br />
|Sashka <br />
|"The Wirtinger Number of a knot equals its bridge number II"<br />
|-<br />
|Feb 24<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation III"<br />
|-<br />
|Mar 3<br />
|Manuel Gonzalez Villa <br />
|"Multiplier ideals of irreducible plane curve singularities"<br />
|-<br />
|}<br />
<br />
== Fall 2016==<br />
Wednesdays at 14:30 VV901<br />
<br />
The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 14 (W)<br />
|Laurentiu Maxim <br />
|"Alexander-type invariants of hypersurface complements"<br />
|-<br />
|Sept. 21 (W)<br />
|Botong Wang <br />
|"Cohomology jump loci"<br />
|-<br />
|Sept. 28 (W)<br />
|Alexandra Kjuchukova <br />
|"On the Bridge Number vs Meridional Rank Conjecture"<br />
|-<br />
|Oct 5 (W)<br />
|Manuel Gonzalez Villa <br />
|"Introduction to Newton polyhedra"<br />
|-<br />
|Oct 12 (W)<br />
|Manuel Gonzalez Villa <br />
|"More on Newton polyhedra"<br />
|-<br />
|Oct 26 (W)<br />
|Christian Geske<br />
|"Intersection Spaces"<br />
|-<br />
|Nov 2 (W)<br />
|Christian Geske<br />
|"Intersection Spaces Continued"<br />
|-<br />
|Nov 9 (W)<br />
|CANCELLED<br />
|-<br />
|Nov 16 (W)<br />
|Eva Elduque<br />
|"Braids and the fundamental group of plane curve complements"<br />
|-<br />
|Nov 30 (W)<br />
|Laurentiu Maxim<br />
|"Novikov homology of hypersurface complements"<br />
|-<br />
|Dec 7 (W)<br />
|CANCELLED<br />
|-<br />
|Dec 14 (W)<br />
|Eva Elduque<br />
|Specialty Exam: "Twisted Alexander invariants of plane curve complements"<br />
|-<br />
|}<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|May. 2 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Eva Elduque or Christian Geske.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=14452Graduate/Postdoc Topology and Singularities Seminar2017-10-26T19:23:32Z<p>Cgeske: /* Fall 2017 */</p>
<hr />
<div>== Fall 2017==<br />
<br />
The Seminar meets at 3:30 to 4:30 pm on Wednesdays in Van Vleck 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Oct 4<br />
|Eva Elduque <br />
|"Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (I)"<br />
|-<br />
|-<br />
|Oct 11<br />
|Eva Elduque <br />
|"Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (II)"<br />
|-<br />
|-<br />
|Oct 18<br />
|Sebastian Baader <br />
|"Dehn twist length in mapping class groups"<br />
|-<br />
|-<br />
|Oct 25<br />
|Cancelled <br />
|-<br />
|-<br />
|Nov 1<br />
|Christian Geske <br />
|"Algebraic Intersection Spaces (I)"<br />
|}<br />
<br />
== Spring 2017==<br />
Fridays at 11:00 VV901<br />
<br />
The Seminar meets on Fridays at 11:00 pm in Van Vleck 901, and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Jan 27<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation I"<br />
|-<br />
|Feb 3<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation II"<br />
|-<br />
|Feb 10<br />
|Sashka <br />
|"The Wirtinger Number of a knot equals its bridge number I"<br />
|-<br />
|Feb 17<br />
|Sashka <br />
|"The Wirtinger Number of a knot equals its bridge number II"<br />
|-<br />
|Feb 24<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation III"<br />
|-<br />
|Mar 3<br />
|Manuel Gonzalez Villa <br />
|"Multiplier ideals of irreducible plane curve singularities"<br />
|-<br />
|}<br />
<br />
== Fall 2016==<br />
Wednesdays at 14:30 VV901<br />
<br />
The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 14 (W)<br />
|Laurentiu Maxim <br />
|"Alexander-type invariants of hypersurface complements"<br />
|-<br />
|Sept. 21 (W)<br />
|Botong Wang <br />
|"Cohomology jump loci"<br />
|-<br />
|Sept. 28 (W)<br />
|Alexandra Kjuchukova <br />
|"On the Bridge Number vs Meridional Rank Conjecture"<br />
|-<br />
|Oct 5 (W)<br />
|Manuel Gonzalez Villa <br />
|"Introduction to Newton polyhedra"<br />
|-<br />
|Oct 12 (W)<br />
|Manuel Gonzalez Villa <br />
|"More on Newton polyhedra"<br />
|-<br />
|Oct 26 (W)<br />
|Christian Geske<br />
|"Intersection Spaces"<br />
|-<br />
|Nov 2 (W)<br />
|Christian Geske<br />
|"Intersection Spaces Continued"<br />
|-<br />
|Nov 9 (W)<br />
|CANCELLED<br />
|-<br />
|Nov 16 (W)<br />
|Eva Elduque<br />
|"Braids and the fundamental group of plane curve complements"<br />
|-<br />
|Nov 30 (W)<br />
|Laurentiu Maxim<br />
|"Novikov homology of hypersurface complements"<br />
|-<br />
|Dec 7 (W)<br />
|CANCELLED<br />
|-<br />
|Dec 14 (W)<br />
|Eva Elduque<br />
|Specialty Exam: "Twisted Alexander invariants of plane curve complements"<br />
|-<br />
|}<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|May. 2 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Eva Elduque or Christian Geske.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=14451Graduate/Postdoc Topology and Singularities Seminar2017-10-26T19:22:56Z<p>Cgeske: /* Fall 2017 */</p>
<hr />
<div>== Fall 2017==<br />
<br />
The Seminar meets at 3:30 to 4:30 pm on Wednesdays in Van Vleck 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Oct 4<br />
|Eva Elduque <br />
|"Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (I)"<br />
|-<br />
|-<br />
|Oct 11<br />
|Eva Elduque <br />
|"Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (II)"<br />
|-<br />
|-<br />
|Oct 18<br />
|Sebastian Baader <br />
|"Dehn twist length in mapping class groups"<br />
|-<br />
|-<br />
|Oct 25<br />
|Cancelled <br />
|-<br />
|-<br />
|Nov 1<br />
|Christian Geske <br />
|"Algebraic Intersection Spaces"<br />
|}<br />
<br />
== Spring 2017==<br />
Fridays at 11:00 VV901<br />
<br />
The Seminar meets on Fridays at 11:00 pm in Van Vleck 901, and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Jan 27<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation I"<br />
|-<br />
|Feb 3<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation II"<br />
|-<br />
|Feb 10<br />
|Sashka <br />
|"The Wirtinger Number of a knot equals its bridge number I"<br />
|-<br />
|Feb 17<br />
|Sashka <br />
|"The Wirtinger Number of a knot equals its bridge number II"<br />
|-<br />
|Feb 24<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation III"<br />
|-<br />
|Mar 3<br />
|Manuel Gonzalez Villa <br />
|"Multiplier ideals of irreducible plane curve singularities"<br />
|-<br />
|}<br />
<br />
== Fall 2016==<br />
Wednesdays at 14:30 VV901<br />
<br />
The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 14 (W)<br />
|Laurentiu Maxim <br />
|"Alexander-type invariants of hypersurface complements"<br />
|-<br />
|Sept. 21 (W)<br />
|Botong Wang <br />
|"Cohomology jump loci"<br />
|-<br />
|Sept. 28 (W)<br />
|Alexandra Kjuchukova <br />
|"On the Bridge Number vs Meridional Rank Conjecture"<br />
|-<br />
|Oct 5 (W)<br />
|Manuel Gonzalez Villa <br />
|"Introduction to Newton polyhedra"<br />
|-<br />
|Oct 12 (W)<br />
|Manuel Gonzalez Villa <br />
|"More on Newton polyhedra"<br />
|-<br />
|Oct 26 (W)<br />
|Christian Geske<br />
|"Intersection Spaces"<br />
|-<br />
|Nov 2 (W)<br />
|Christian Geske<br />
|"Intersection Spaces Continued"<br />
|-<br />
|Nov 9 (W)<br />
|CANCELLED<br />
|-<br />
|Nov 16 (W)<br />
|Eva Elduque<br />
|"Braids and the fundamental group of plane curve complements"<br />
|-<br />
|Nov 30 (W)<br />
|Laurentiu Maxim<br />
|"Novikov homology of hypersurface complements"<br />
|-<br />
|Dec 7 (W)<br />
|CANCELLED<br />
|-<br />
|Dec 14 (W)<br />
|Eva Elduque<br />
|Specialty Exam: "Twisted Alexander invariants of plane curve complements"<br />
|-<br />
|}<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|May. 2 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Eva Elduque or Christian Geske.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=14382Graduate/Postdoc Topology and Singularities Seminar2017-10-17T16:34:57Z<p>Cgeske: /* Fall 2017 */</p>
<hr />
<div>== Fall 2017==<br />
<br />
The Seminar meets at 3:30 to 4:30 pm on Wednesdays in Van Vleck 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Oct 4<br />
|Eva Elduque <br />
|"Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (I)"<br />
|-<br />
|-<br />
|Oct 11<br />
|Eva Elduque <br />
|"Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (II)"<br />
|-<br />
|-<br />
|Oct 18<br />
|Sebastian Baader <br />
|"Dehn twist length in mapping class groups"<br />
|-<br />
|-<br />
|Oct 25<br />
|Cancelled <br />
|}<br />
<br />
== Spring 2017==<br />
Fridays at 11:00 VV901<br />
<br />
The Seminar meets on Fridays at 11:00 pm in Van Vleck 901, and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Jan 27<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation I"<br />
|-<br />
|Feb 3<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation II"<br />
|-<br />
|Feb 10<br />
|Sashka <br />
|"The Wirtinger Number of a knot equals its bridge number I"<br />
|-<br />
|Feb 17<br />
|Sashka <br />
|"The Wirtinger Number of a knot equals its bridge number II"<br />
|-<br />
|Feb 24<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation III"<br />
|-<br />
|Mar 3<br />
|Manuel Gonzalez Villa <br />
|"Multiplier ideals of irreducible plane curve singularities"<br />
|-<br />
|}<br />
<br />
== Fall 2016==<br />
Wednesdays at 14:30 VV901<br />
<br />
The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 14 (W)<br />
|Laurentiu Maxim <br />
|"Alexander-type invariants of hypersurface complements"<br />
|-<br />
|Sept. 21 (W)<br />
|Botong Wang <br />
|"Cohomology jump loci"<br />
|-<br />
|Sept. 28 (W)<br />
|Alexandra Kjuchukova <br />
|"On the Bridge Number vs Meridional Rank Conjecture"<br />
|-<br />
|Oct 5 (W)<br />
|Manuel Gonzalez Villa <br />
|"Introduction to Newton polyhedra"<br />
|-<br />
|Oct 12 (W)<br />
|Manuel Gonzalez Villa <br />
|"More on Newton polyhedra"<br />
|-<br />
|Oct 26 (W)<br />
|Christian Geske<br />
|"Intersection Spaces"<br />
|-<br />
|Nov 2 (W)<br />
|Christian Geske<br />
|"Intersection Spaces Continued"<br />
|-<br />
|Nov 9 (W)<br />
|CANCELLED<br />
|-<br />
|Nov 16 (W)<br />
|Eva Elduque<br />
|"Braids and the fundamental group of plane curve complements"<br />
|-<br />
|Nov 30 (W)<br />
|Laurentiu Maxim<br />
|"Novikov homology of hypersurface complements"<br />
|-<br />
|Dec 7 (W)<br />
|CANCELLED<br />
|-<br />
|Dec 14 (W)<br />
|Eva Elduque<br />
|Specialty Exam: "Twisted Alexander invariants of plane curve complements"<br />
|-<br />
|}<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|May. 2 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Eva Elduque or Christian Geske.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=13408Graduate/Postdoc Topology and Singularities Seminar2017-02-22T19:32:31Z<p>Cgeske: /* Spring 2017 */</p>
<hr />
<div>== Spring 2017==<br />
Fridays at 11:00 VV901<br />
<br />
The Seminar meets on Fridays at 11:00 pm in Van Vleck 901, and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Jan 27<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation I"<br />
|-<br />
|Feb 3<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation II"<br />
|-<br />
|Feb 10<br />
|Sashka <br />
|"The Wirtinger Number of a knot equals its bridge number I"<br />
|-<br />
|Feb 17<br />
|Sashka <br />
|"The Wirtinger Number of a knot equals its bridge number II"<br />
|-<br />
|Feb 24<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation III"<br />
|-<br />
|}<br />
<br />
== Fall 2016==<br />
Wednesdays at 14:30 VV901<br />
<br />
The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 14 (W)<br />
|Laurentiu Maxim <br />
|"Alexander-type invariants of hypersurface complements"<br />
|-<br />
|Sept. 21 (W)<br />
|Botong Wang <br />
|"Cohomology jump loci"<br />
|-<br />
|Sept. 28 (W)<br />
|Alexandra Kjuchukova <br />
|"On the Bridge Number vs Meridional Rank Conjecture"<br />
|-<br />
|Oct 5 (W)<br />
|Manuel Gonzalez Villa <br />
|"Introduction to Newton polyhedra"<br />
|-<br />
|Oct 12 (W)<br />
|Manuel Gonzalez Villa <br />
|"More on Newton polyhedra"<br />
|-<br />
|Oct 26 (W)<br />
|Christian Geske<br />
|"Intersection Spaces"<br />
|-<br />
|Nov 2 (W)<br />
|Christian Geske<br />
|"Intersection Spaces Continued"<br />
|-<br />
|Nov 9 (W)<br />
|CANCELLED<br />
|-<br />
|Nov 16 (W)<br />
|Eva Elduque<br />
|"Braids and the fundamental group of plane curve complements"<br />
|-<br />
|Nov 30 (W)<br />
|Laurentiu Maxim<br />
|"Novikov homology of hypersurface complements"<br />
|-<br />
|Dec 7 (W)<br />
|CANCELLED<br />
|-<br />
|Dec 14 (W)<br />
|Eva Elduque<br />
|Specialty Exam: "Twisted Alexander invariants of plane curve complements"<br />
|-<br />
|}<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|May. 2 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Eva Elduque or Christian Geske.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=13047Graduate/Postdoc Topology and Singularities Seminar2017-01-19T15:04:35Z<p>Cgeske: </p>
<hr />
<div>== Spring 2017==<br />
Wednesdays at 14:30 VV901<br />
<br />
The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901, and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Jan. 25<br />
|Christian Geske <br />
|"Intersection Spaces and Equivariant Moore Approximation I"<br />
|-<br />
|}<br />
<br />
== Fall 2016==<br />
Wednesdays at 14:30 VV901<br />
<br />
The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 14 (W)<br />
|Laurentiu Maxim <br />
|"Alexander-type invariants of hypersurface complements"<br />
|-<br />
|Sept. 21 (W)<br />
|Botong Wang <br />
|"Cohomology jump loci"<br />
|-<br />
|Sept. 28 (W)<br />
|Alexandra Kjuchukova <br />
|"On the Bridge Number vs Meridional Rank Conjecture"<br />
|-<br />
|Oct 5 (W)<br />
|Manuel Gonzalez Villa <br />
|"Introduction to Newton polyhedra"<br />
|-<br />
|Oct 12 (W)<br />
|Manuel Gonzalez Villa <br />
|"More on Newton polyhedra"<br />
|-<br />
|Oct 26 (W)<br />
|Christian Geske<br />
|"Intersection Spaces"<br />
|-<br />
|Nov 2 (W)<br />
|Christian Geske<br />
|"Intersection Spaces Continued"<br />
|-<br />
|Nov 9 (W)<br />
|CANCELLED<br />
|-<br />
|Nov 16 (W)<br />
|Eva Elduque<br />
|"Braids and the fundamental group of plane curve complements"<br />
|-<br />
|Nov 30 (W)<br />
|Laurentiu Maxim<br />
|"Novikov homology of hypersurface complements"<br />
|-<br />
|Dec 7 (W)<br />
|CANCELLED<br />
|-<br />
|Dec 14 (W)<br />
|Eva Elduque<br />
|Specialty Exam: "Twisted Alexander invariants of plane curve complements"<br />
|-<br />
|}<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|May. 2 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Eva Elduque or Christian Geske.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=12620Graduate/Postdoc Topology and Singularities Seminar2016-10-26T20:58:43Z<p>Cgeske: </p>
<hr />
<div>== Fall 2016==<br />
Wednesdays at 14:30 VV901<br />
<br />
The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 14 (W)<br />
|Laurentiu Maxim <br />
|"Alexander-type invariants of hypersurface complements"<br />
|-<br />
|Sept. 21 (W)<br />
|Botong Wang <br />
|"Cohomology jump loci"<br />
|-<br />
|Sept. 28 (W)<br />
|Alexandra Kjuchukova <br />
|"On the Bridge Number vs Meridional Rank Conjecture"<br />
|-<br />
|Oct 5 (W)<br />
|Manuel Gonzalez Villa <br />
|"Introduction to Newton polyhedra"<br />
|-<br />
|Oct 12 (W)<br />
|Manuel Gonzalez Villa <br />
|"More on Newton polyhedra"<br />
|-<br />
|Oct 26 (W)<br />
|Christian Geske<br />
|"Intersection Spaces"<br />
|-<br />
|Nov 2 (W)<br />
|Christian Geske<br />
|"Intersection Spaces Continued"<br />
|-<br />
|}<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|May. 2 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Eva Elduque or Christian Geske.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=12496Graduate/Postdoc Topology and Singularities Seminar2016-10-14T17:06:01Z<p>Cgeske: </p>
<hr />
<div>== Fall 2016==<br />
Wednesdays at 14:30 VV901<br />
<br />
The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 14 (W)<br />
|Laurentiu Maxim <br />
|"Alexander-type invariants of hypersurface complements"<br />
|-<br />
|Sept. 21 (W)<br />
|Botong Wang <br />
|"Cohomology jump loci"<br />
|-<br />
|Sept. 28 (W)<br />
|Alexandra Kjuchukova <br />
|"On the Bridge Number vs Meridional Rank Conjecture"<br />
|-<br />
|Oct 5 (W)<br />
|Manuel Gonzalez Villa <br />
|"Introduction to Newton polyhedra"<br />
|-<br />
|Oct 12 (W)<br />
|Manuel Gonzalez Villa <br />
|"More on Newton polyhedra"<br />
|-<br />
|Oct 26 (W)<br />
|Christian Geske<br />
|"Intersection Spaces"<br />
|-<br />
|}<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|May. 2 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Eva Elduque or Christian Geske.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=11897Graduate/Postdoc Topology and Singularities Seminar2016-05-02T19:11:17Z<p>Cgeske: </p>
<hr />
<div>The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.<br />
<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|May. 2 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Eva Elduque or Christian Geske.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=11859Graduate/Postdoc Topology and Singularities Seminar2016-04-20T21:23:42Z<p>Cgeske: </p>
<hr />
<div>The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.<br />
<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|TBA<br />
|-<br />
|May. 2 (M)<br />
|<br />
|TBA<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Eva Elduque or Christian Geske.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=11762Graduate/Postdoc Topology and Singularities Seminar2016-04-07T19:30:50Z<p>Cgeske: /* Spring 2016 */</p>
<hr />
<div>The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Tommy Wong. <br />
<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont.<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|TBA<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|TBA<br />
|-<br />
|May. 2 (M)<br />
|<br />
|TBA<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Tommy Wong.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=11761Graduate/Postdoc Topology and Singularities Seminar2016-04-07T19:30:24Z<p>Cgeske: /* Spring 2016 */</p>
<hr />
<div>The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Tommy Wong. <br />
<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 cont...<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|TBA<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|TBA<br />
|-<br />
|May. 2 (M)<br />
|<br />
|TBA<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Tommy Wong.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=11760Graduate/Postdoc Topology and Singularities Seminar2016-04-07T19:29:55Z<p>Cgeske: /* Spring 2016 */</p>
<hr />
<div>The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Tommy Wong. <br />
<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|Eva Elduque<br />
|Section 2.3: ''Direct and inverse image''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Cancelled <br />
|-<br />
|Apr. 4 (M)<br />
|<br />
|Cancelled<br />
|-<br />
|Apr. 11 (M)<br />
|Christian Geske<br />
|Section 2.3 Continued<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|TBA<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|TBA<br />
|-<br />
|May. 2 (M)<br />
|<br />
|TBA<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Tommy Wong.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeskehttps://wiki.math.wisc.edu/index.php?title=Graduate/Postdoc_Topology_and_Singularities_Seminar&diff=11560Graduate/Postdoc Topology and Singularities Seminar2016-02-27T00:04:07Z<p>Cgeske: /* Spring 2016 */</p>
<hr />
<div>The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.<br />
<br />
The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Tommy Wong. <br />
<br />
<br />
== Spring 2016==<br />
Mondays at 3:20 B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 8 (M)<br />
|Christian Geske<br />
|Section 1.1 and 1.2: ''Category of complexes and Homotopical category''<br />
|-<br />
|Feb. 15 (M)<br />
|Eva Elduque<br />
|Sections 1.3 and 1.4: ''Derived category and derived functors''<br />
|-<br />
|Feb. 22 (M)<br />
|Botong Wang<br />
|Sections 2.1 and 2.2: ''Generalities on Sheaves and Derived tensor products''<br />
|-<br />
|Feb. 29 (M)<br />
|Christian Geske<br />
|''Hypercohomology and Holomorphic Differential Forms on Analytic Varieties''<br />
|-<br />
|Mar. 7 (M)<br />
|<br />
|Sections 2.3 and 2.4: ''Direct and inverse image and The adjunction triangle''<br />
|-<br />
|Mar. 14 (M)<br />
|<br />
|Section 2.5: ''Local Systems''<br />
|-<br />
|Mar. 28 (M)<br />
|<br />
|Section 3.1 and 3.2: ''Cohomological dimensions and the functor f!''<br />
|-<br />
|Apr.. 4 (M)<br />
|<br />
|Sections 3.3 and 3.4: ''Poincare and Alexander duality and Vanishing results''<br />
|-<br />
|Apr. 11 (M)<br />
|<br />
|Section 4.1: ''Constructible sheaves''<br />
|-<br />
|Apr. 18 (M)<br />
|<br />
|Section 4.2 and 4.3: ''Nearby and vanishing cycles and Characteristic varieties and cycles''<br />
|-<br />
|Apr. 25 (M)<br />
|<br />
|Section 5.1 and 5.2: ''Perverse sheaves and their properties''<br />
|-<br />
|May. 2 (M)<br />
|<br />
|Section 5.3: ''D-modules''<br />
|-<br />
|}<br />
<br />
If you would like to present a topic, please contact Tommy Wong.<br />
<br />
== Abstracts ==<br />
<br />
<br />
(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).<br />
<br />
This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).<br />
<br />
The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.<br />
<br />
Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.<br />
<br />
== Fall 2015 ==<br />
<br />
Thursdays 4pm in B139VV<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 24 (Th)<br />
|KaiHo (Tommy) Wong<br />
|''Twisted Alexander Invariant for Knots and Plane Curves''<br />
|-<br />
|Oct. 1 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers I''<br />
|-<br />
|Oct. 8 (Th)<br />
|Alexandra (Sashka) Kjuchukova<br />
|''Linking numbers and branched covers II''<br />
|-<br />
|Oct. 15 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture I''<br />
|-<br />
|Oct. 22 (Th)<br />
|Yun Su (Suky)<br />
|Pretalk ''Higher-order degrees of hypersurface complements.'', Survey on Alexander polynomial for plane curves.<br />
|-<br />
|Oct. 29 (Th)<br />
|Yun Su (Suky)<br />
|Aftertalk ''Higher-order degrees of hypersurface complements.''<br />
|-<br />
|Nov. 5 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture II''<br />
|-<br />
|Nov. 12 (Th)<br />
|Manuel Gonzalez Villa<br />
|''On poles of zeta functions and monodromy conjecture III''<br />
|-<br />
|Nov. 19 (Th)<br />
|Eva Elduque<br />
|''Stiefel-Whitney classes''<br />
|-<br />
|Dec. 3 (Th)<br />
|Eva Elduque<br />
|''Grass-mania!''<br />
|-<br />
|Dec. 10 (Th)<br />
|KaiHo (Tommy) Wong<br />
|Pretalk ''Milnor Fiber of Complex Hyperplane Arrangements''<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Th, Sep 24: Tommy===<br />
Twisted Alexander Invariant of Knots and Plane Curves.<br />
<br />
I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.<br />
<br />
<br />
===Th, Oct 1 and 8: Sashka===<br />
Linking numbers and branched coverings I and II<br />
<br />
Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves. <br />
<br />
In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.<br />
<br />
===Th, Oct 15, Nov 5 and Nov 12: Manuel===<br />
On poles of zeta functions and monodromy conjecture I and II<br />
<br />
Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.<br />
<br />
===Th, Nov 19: Eva===<br />
Stiefel-Whitney classes<br />
<br />
Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.<br />
<br />
===Th, Dec 3: Eva===<br />
Grass-mania!<br />
<br />
In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.<br />
<br />
This talk is for the most part self contained, so it doesn't matter if you missed the previous one.<br />
<br />
<br />
===Th, Dec 10: Tommy===<br />
<br />
A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!<br />
<br />
I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.<br />
<br />
== Spring 2014 ==<br />
<br />
We continue with Professor Alex Suciu's work.<br />
<br />
== Fall 2014 ==<br />
<br />
We follow Professor Alex Suciu's work this semester.<br />
<br />
http://www.northeastern.edu/suciu/publications.html<br />
<br />
But we will not meet at a regular basis.<br />
<br />
<br />
== Spring 2014 ==<br />
<br />
We meet on Tuesdays 3:30-4:25pm in room B211.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 25 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition I''<br />
|-<br />
|Mar. 4 (Tue)<br />
|Yongqiang Liu<br />
|''Monodromy Decomposition II''<br />
|-<br />
|Mar. 25 (Tue)<br />
|KaiHo Wong<br />
|''Conjecture of lower bounds of Alexander polynomial''<br />
|-<br />
|Apr. 8 (Tue)<br />
|Yongqiang Liu<br />
|''Nearby Cycles and Alexander Modules''<br />
|-<br />
|}<br />
<br />
== Fall 2013 ==<br />
<br />
We are learning Hodge Theory this semester and will be following three books:<br />
<br />
1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II<br />
<br />
2. Peters, Steenbrink, Mixed Hodge Structures <br />
<br />
We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sep. 18 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Sep. 25 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor Fibration at infinity of polynomial map''<br />
|-<br />
|Oct. 9 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|-<br />
|Oct. 16 (Wed)<br />
|Yongqiang Liu<br />
|''Polynomial singularities''<br />
|-<br />
|Nov. 13 (Wed)<br />
|KaiHo Wong<br />
|Discussions on book material<br />
|}<br />
<br />
== Spring 2013 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Feb. 6 (Wed)<br />
|Jeff Poskin<br />
|''Toric Varieties III''<br />
|-<br />
|Feb.13 (Wed)<br />
|Yongqiang Liu<br />
|''Intersection Alexander Module''<br />
|-<br />
|Feb.20 (Wed)<br />
|Yun Su (Suky)<br />
|''How do singularities change shape and view of objects?''<br />
|-<br />
|Feb.27 (Wed)<br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements''<br />
|-<br />
|Mar.20 (Wed)<br />
|J&ouml;rg Sch&uuml;rmann (University of M&uuml;nster, Germany)<br />
|''Characteristic classes of singular toric varieties''<br />
|-<br />
|Apr. 3 (Wed) <br />
|KaiHo Wong<br />
|''Fundamental groups of plane curves complements II''<br />
|-<br />
|Apr.10 (Wed)<br />
|Yongqiang Liu<br />
|''Milnor fiber of local function germ''<br />
|-<br />
|Apr.17 (Wed) 2:45pm-3:45pm (Note the different time)<br />
|KaiHo Wong<br />
|''Formula of Alexander polynomials of plane curves''<br />
|-<br />
|-<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Wed, 2/27: Tommy===<br />
''Fundamental groups of plane curves complements''<br />
<br />
I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed. <br />
<br />
<br />
<br />
<br />
== Fall 2012 ==<br />
<br />
{| cellpadding="5"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
|-<br />
|Sept. 18 (Tue)<br />
|KaiHo Wong <br />
|Organization and ''Milnor fibration and Milnor Fiber''<br />
|-<br />
|Sept. 25 (Tue)<br />
|KaiHo Wong <br />
|''Algebraic links and exotic spheres''<br />
|-<br />
|Oct. 4 (Thu)<br />
|Yun Su (Suky)<br />
|''Alexander polynomial of complex algebraic curve'' (Note the different day but same time and location)<br />
|-<br />
|Oct. 11 (Thu)<br />
|Yongqiang Liu<br />
|''Sheaves and Hypercohomology''<br />
|-<br />
|Oct. 18 (Thu)<br />
|Jeff Poskin<br />
|''Toric Varieties II''<br />
|-<br />
|Nov. 1 (Thu)<br />
|Yongqiang Liu<br />
|''Mixed Hodge Structure''<br />
|-<br />
|Nov. 15 (Thu)<br />
|KaiHo Wong<br />
|''Euler characteristics of hypersurfaces with isolated singularities''<br />
|-<br />
|Nov. 29 (Thu)<br />
|Markus Banagl, University of Heidelberg<br />
|''High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres''<br />
|-<br />
|}<br />
== Abstracts ==<br />
<br />
===Thu, 10/4: Suky===<br />
''Alexander polynomial of complex algebraic curve''<br />
<br />
I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. <br />
From the definition, it is clear that Alexander polynomial is an topological invariant for curves.<br />
I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. <br />
Calculations of some examples will be provided.</div>Cgeske