Graduate Algebraic Geometry Seminar Spring 2022: Difference between revisions

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'''
'''
'''When:''' TBD
'''When:''' 4:30-5:30 PM Thursdays


'''Where:''' TBD
'''Where:''' VV B231
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]


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'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is [https://groups.google.com/u/2/a/g-groups.wisc.edu/g/gags here].
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is [https://groups.google.com/u/2/a/g-groups.wisc.edu/g/gags here].


''' Organizers: ''' [https://johndcobb.github.io John Cobb], [https://sites.google.com/view/colincrowley/home Colin Crowley].
''' Organizers: ''' [https://johndcobb.github.io John Cobb], Yu (Joey) Luo


== Give a talk! ==
== Give a talk! ==
We need volunteers to give talks this semester. If you're interested, please fill out [https://forms.gle/sa3ARndYSkBhT6LR9 this form]. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the [https://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_Algebraic_Geometry_Seminar main page].
We need volunteers to give talks this semester. If you're interested, please fill out [https://forms.gle/iwvCQPKp3mDD3HZd9 this form]. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the [https://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_Algebraic_Geometry_Seminar main page].


=== Spring 2022 Topic Wish List ===
=== Spring 2022 Topic Wish List ===
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.
* Computing things about Toric varieties
* Hilbert Schemes
* Reductive groups and flag varieties
* Reductive groups and flag varieties
* Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
* Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
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| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''
| bgcolor="#BCD2EE" width="300" align="center"|'''Title'''
|-
|-
| bgcolor="#E0E0E0"| February 2
| bgcolor="#E0E0E0"| February 10
| bgcolor="#C6D46E"|  
| bgcolor="#C6D46E"| Everyone
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| ]]
| bgcolor="#BCE2FE"|[[#February 10| Informal chat session ]]
|-
|-
| bgcolor="#E0E0E0"| February 5
| bgcolor="#E0E0E0"| February 17
| bgcolor="#C6D46E"| Asvin Gothandaraman
| bgcolor="#C6D46E"| Asvin G
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]
| bgcolor="#BCE2FE"|[[#February 17| Motives ]]
|-
|-
| bgcolor="#E0E0E0"| February 12
| bgcolor="#E0E0E0"| February 24
| bgcolor="#C6D46E"| Qiao He
| bgcolor="#C6D46E"| Yu Luo
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]
| bgcolor="#BCE2FE"|[[#February 24| Riemann-Hilbert Correspondence ]]
|-
|-
| bgcolor="#E0E0E0"| February 19
| bgcolor="#E0E0E0"| March 10
| bgcolor="#C6D46E"| Dima Arinkin
| bgcolor="#C6D46E"| Colin Crowley
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]
| bgcolor="#BCE2FE"|[[#March 10| An introduction to Tropicalization ]]
|-
|-
| bgcolor="#E0E0E0"| February 26
| bgcolor="#E0E0E0"| March 31
| bgcolor="#C6D46E"| Connor Simpson
| bgcolor="#C6D46E"| Ruofan
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]
| bgcolor="#BCE2FE"|[[#March 31| Motivic class of stack of finite modules over a cusp ]]
|-
|-
| bgcolor="#E0E0E0"| March 4
| bgcolor="#E0E0E0"| April 7
| bgcolor="#C6D46E"| Peter
| bgcolor="#C6D46E"| Alex Hof
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]
| bgcolor="#BCE2FE"|[[#April 7| Geometric Intuitions for Flatness]]
|-
|-
| bgcolor="#E0E0E0"| March 11
| bgcolor="#E0E0E0"| April 14
| bgcolor="#C6D46E"| Caitlyn Booms
| bgcolor="#C6D46E"| Caitlyn Booms
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]
| bgcolor="#BCE2FE"|[[#April 14| Virtual criterion for generalized Eagon-Northcott complexes ]]
|-
|-
| bgcolor="#E0E0E0"| March 25
| bgcolor="#E0E0E0"| April 21
| bgcolor="#C6D46E"| Steven He
| bgcolor="#C6D46E"| Connor Simpson
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Braid group action on derived categories]]
| bgcolor="#BCE2FE"|[[#April 21| Symplectic geometry and invariant theory ]]
|-
|-
| bgcolor="#E0E0E0"| April 1
| bgcolor="#E0E0E0"| April 28
| bgcolor="#C6D46E"| Vlad Sotirov
| bgcolor="#C6D46E"| Karan
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]
| bgcolor="#BCE2FE"|[[#April 28| Using varieties to study polynomial neural networks ]]
|-
|-
| bgcolor="#E0E0E0"| April 8
| bgcolor="#E0E0E0"| May 5
| bgcolor="#C6D46E"| Maya Banks
| bgcolor="#C6D46E"| Ellie Thieu
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]
| bgcolor="#BCE2FE"|[[#May 5| Visualizing Cohomology ]]
|-
| bgcolor="#E0E0E0"| April 15
| bgcolor="#C6D46E"| Alex Hof
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]
|-
| bgcolor="#E0E0E0"| April 22
| bgcolor="#C6D46E"| Ruofan
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Birational geometry: existence of rational curves]]
|-
| bgcolor="#E0E0E0"| April 29
| bgcolor="#C6D46E"| John Cobb
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]
|}
|}
</center>
</center>


=== January 29 ===
=== February 10 ===
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Everyone '''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Lefschetz hyperplane section theorem via Morse theory
| bgcolor="#BCD2EE"  align="center" | Title: Informal chat session
|-
|-
| bgcolor="#BCD2EE"  | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.
| bgcolor="#BCD2EE"  | Abstract: Bring your questions!
|}                                                                         
|}                                                                         
</center>
</center>


=== February 5 ===
=== February 17 ===
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin G '''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: An introduction to unirationality
| bgcolor="#BCD2EE"  align="center" | Title: Motives
|-
|-
| bgcolor="#BCD2EE"  | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected.
| bgcolor="#BCD2EE"  | Abstract: Some motivation behind motives
|}                                                                         
|}                                                                         
</center>
</center>


=== February 12 ===
=== February 24 ===
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Yu LUO (Joey) '''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title:  
| bgcolor="#BCD2EE"  align="center" | Title: Riemann-Hilbert Correspondence
|-
|-
| bgcolor="#BCD2EE"  | Abstract:  
| bgcolor="#BCD2EE"  | Abstract: During the talk, I will start with "nonsingular" version of Riemann-Hilbert correspondence between flat vector bundles and local systems. Then I will introduce the regular singularity, then sketch the Riemann-Hilbert correspondence with regular singularity. If time permit, I will brief mention some applications.
\end{abstract}
|}                                                                         
|}                                                                         
</center>
</center>


=== February 19 ===
=== March 10 ===
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Colin Crowley '''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Blowing down, blowing up: surface geometry
| bgcolor="#BCD2EE"  align="center" | Title: An introduction to Tropicalization
|-
|-
| bgcolor="#BCD2EE"  | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?
| bgcolor="#BCD2EE"  | Abstract: Tropicalization is a logarithmic process (functor) that takes embedded algebraic varieties to polyhedral complexes. The complexes that are in the image have some additional structure which leads to the definition of a tropical variety, the main object of study in tropical geometry. I'll talk about the first paper to use these ideas, and the problem that they were used to solve.
 
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.
 
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.)
|}                                                                         
|}                                                                         
</center>
</center>


=== February 26 ===
=== March 31 ===
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Ruofan '''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Intro to Toric Varieties
| bgcolor="#BCD2EE"  align="center" | Title: Motivic class of stack of finite modules over a cusp
|-
|-
| bgcolor="#BCD2EE"  | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.
| bgcolor="#BCD2EE"  | Abstract: We find the explicit motivic class of stack of finite modules over some complete local rings. This is some recent work originated from a project with Asvin and Yifan.
|}                                                                         
|}                                                                         
</center>
</center>


=== March 4 ===
=== April 7 ===
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Alex Hof '''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem
| bgcolor="#BCD2EE"  align="center" | Title: Geometric Intuitions for Flatness
|-
|-
| bgcolor="#BCD2EE"  | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.
| bgcolor="#BCD2EE"  | Abstract: Flatness is often described as the correct characterization of what it means to have "a nicely varying family of things" in the setting of algebraic geometry. In this talk, which is intended to be pretty low-key, I'll dig a little bit more into what that means, and discuss some theorems and examples that refine and clarify this intuition in various ways.
|}                                                                         
|}                                                                         
</center>
</center>


=== March 11 ===
=== April 14 ===
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Caitlyn Booms '''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Intro to Stanley-Reisner Theory
| bgcolor="#BCD2EE"  align="center" | Title: Virtual criterion for generalized Eagon-Northcott complexes
|-
|-
| bgcolor="#BCD2EE"  | Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.
| bgcolor="#BCD2EE"  | Abstract: The Eagon-Northcott complex of a map of finitely generated free modules has been an interest of study since 1962, as it generically resolves the ideal of maximal minors of the matrix that defines the map. In 1975, Buchsbaum and Eisenbud described a family of generalized Eagon-Northcott complexes associated to a map of free modules, which are also generically minimal free resolutions. As introduced by Berkesch, Erman, and Smith in 2020, when working over a smooth projective toric variety, virtual resolutions, rather than minimal free resolutions, are a better tool for understanding the geometry of a space. I will describe sufficient criteria for the family of generalized Eagon-Northcott complexes of a map to be virtual resolutions, thus adding to the known examples of virtual resolutions, particularly those not coming from minimal free resolutions.
|}                                                                         
|}                                                                         
</center>
</center>


=== March 25 ===
=== April 21 ===
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Connor Simpson '''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Braid group action on derived category
| bgcolor="#BCD2EE"  align="center" | Title: Symplectic geometry and invariant theory
|-
|-
| bgcolor="#BCD2EE"  | Abstract: In this talk, I will define spherical object and A_m-configuration in derived category of coherent sheaves, and say a few words about the motivation coming from the homological mirror symmetry.
| bgcolor="#BCD2EE"  | Abstract: We discuss connections between symplectic geometry and invariant theory.
|}                                                                         
|}                                                                         
</center>
</center>


=== April 1 ===
=== April 28 ===
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Karan '''
|-
| bgcolor="#BCD2EE"  align="center" | Title:
|-
|-
| bgcolor="#BCD2EE"  | Abstract:  
| bgcolor="#BCD2EE"  align="center" | Title: Using varieties to study polynomial neural networks
|}                                                                       
</center>


=== April 8 ===
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''
| bgcolor="#BCD2EE"  | Abstract: In this talk, I will exposit the work of Kileel, Trager, and Bruna in their 2019 paper "On the Expressive power of Polynomial Neural Networks". We will look at 1) what a polynomial neural network is and how we can interpret the output such networks as varieties, 2) why the dimension of this variety and the expressive power of this network are related, and 3) how the study of these varieties might tell us something about the architecture of the network.  
|-
| bgcolor="#BCD2EE"  align="center" | Title:
|-
| bgcolor="#BCD2EE"  | Abstract:
|}                                                                       
</center>
 
=== April 15 ===
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory
|-
| bgcolor="#BCD2EE"  | Abstract: Early on in the semester, Colin told us a bit about Morse
Theory, and how it lets us get a handle on the (classical) topology of
smooth complex varieties. As we all know, however, not everything in
life goes smoothly, and so too in algebraic geometry. Singular
varieties, when given the classical topology, are not manifolds, but
they can be described in terms of manifolds by means of something called
a Whitney stratification. This allows us to develop a version of Morse
Theory that applies to singular spaces (and also, with a bit of work, to
smooth spaces that fail to be nice in other ways, like non-compact
manifolds!), called Stratified Morse Theory. After going through the
appropriate definitions and briefly reviewing the results of classical
Morse Theory, we'll discuss the so-called Main Theorem of Stratified
Morse Theory and survey some of its consequences.
|}                                                                         
|}                                                                         
</center>
</center>


=== April 22 ===
=== May 5 ===
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | ''' Ellie Thieu '''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Birational geometry: existence of rational curves
| bgcolor="#BCD2EE"  align="center" | Title: Visualizing Cohomology
|-
|-
| bgcolor="#BCD2EE"  | Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight.  
| bgcolor="#BCD2EE"  | Abstract: We will go through Ravi’s picture book together. While thinking about how to present it, I was faced with the choice of either redrawing the whole picture book, or just be honest about my cheating and use the author’s very own beautiful illustrations. This way of looking at cohomology is not perfect, but it offers a very simple understanding of taking cohomology and spectral sequences. I will deliver how to visualize cohomology, and declare it an early victory. Then we will go as far as time allow to understand spectral sequences.
|}                                                                       
</center>


=== April 29 ===
Please bring your laptop, or anything you can use to follow the illustrations. Because, alas, I will not redraw them on the board.
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''
|-
| bgcolor="#BCD2EE"  align="center" | Title:
|-
| bgcolor="#BCD2EE"  | Abstract:
|}                                                                         
|}                                                                         
</center>
</center>

Latest revision as of 17:33, 3 May 2022

When: 4:30-5:30 PM Thursdays

Where: VV B231

Lizzie the OFFICIAL mascot of GAGS!!

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

Why: The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.

How: If you want to get emails regarding time, place, and talk topics (which are often assigned quite last minute) add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is here.

Organizers: John Cobb, Yu (Joey) Luo

Give a talk!

We need volunteers to give talks this semester. If you're interested, please fill out this form. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the main page.

Spring 2022 Topic Wish List

This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.

  • Hilbert Schemes
  • Reductive groups and flag varieties
  • Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
  • Going from line bundles and divisors to vector bundles and chern classes
  • A History of the Weil Conjectures
  • Mumford & Bayer, "What can be computed in Algebraic Geometry?"
  • A pre talk for any other upcoming talk

Being an audience member

The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:

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

Talks

Date Speaker Title
February 10 Everyone Informal chat session
February 17 Asvin G Motives
February 24 Yu Luo Riemann-Hilbert Correspondence
March 10 Colin Crowley An introduction to Tropicalization
March 31 Ruofan Motivic class of stack of finite modules over a cusp
April 7 Alex Hof Geometric Intuitions for Flatness
April 14 Caitlyn Booms Virtual criterion for generalized Eagon-Northcott complexes
April 21 Connor Simpson Symplectic geometry and invariant theory
April 28 Karan Using varieties to study polynomial neural networks
May 5 Ellie Thieu Visualizing Cohomology

February 10

Everyone
Title: Informal chat session
Abstract: Bring your questions!

February 17

Asvin G
Title: Motives
Abstract: Some motivation behind motives

February 24

Yu LUO (Joey)
Title: Riemann-Hilbert Correspondence
Abstract: During the talk, I will start with "nonsingular" version of Riemann-Hilbert correspondence between flat vector bundles and local systems. Then I will introduce the regular singularity, then sketch the Riemann-Hilbert correspondence with regular singularity. If time permit, I will brief mention some applications.

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March 10

Colin Crowley
Title: An introduction to Tropicalization
Abstract: Tropicalization is a logarithmic process (functor) that takes embedded algebraic varieties to polyhedral complexes. The complexes that are in the image have some additional structure which leads to the definition of a tropical variety, the main object of study in tropical geometry. I'll talk about the first paper to use these ideas, and the problem that they were used to solve.

March 31

Ruofan
Title: Motivic class of stack of finite modules over a cusp
Abstract: We find the explicit motivic class of stack of finite modules over some complete local rings. This is some recent work originated from a project with Asvin and Yifan.

April 7

Alex Hof
Title: Geometric Intuitions for Flatness
Abstract: Flatness is often described as the correct characterization of what it means to have "a nicely varying family of things" in the setting of algebraic geometry. In this talk, which is intended to be pretty low-key, I'll dig a little bit more into what that means, and discuss some theorems and examples that refine and clarify this intuition in various ways.

April 14

Caitlyn Booms
Title: Virtual criterion for generalized Eagon-Northcott complexes
Abstract: The Eagon-Northcott complex of a map of finitely generated free modules has been an interest of study since 1962, as it generically resolves the ideal of maximal minors of the matrix that defines the map. In 1975, Buchsbaum and Eisenbud described a family of generalized Eagon-Northcott complexes associated to a map of free modules, which are also generically minimal free resolutions. As introduced by Berkesch, Erman, and Smith in 2020, when working over a smooth projective toric variety, virtual resolutions, rather than minimal free resolutions, are a better tool for understanding the geometry of a space. I will describe sufficient criteria for the family of generalized Eagon-Northcott complexes of a map to be virtual resolutions, thus adding to the known examples of virtual resolutions, particularly those not coming from minimal free resolutions.

April 21

Connor Simpson
Title: Symplectic geometry and invariant theory
Abstract: We discuss connections between symplectic geometry and invariant theory.

April 28

Karan
Title: Using varieties to study polynomial neural networks
Abstract: In this talk, I will exposit the work of Kileel, Trager, and Bruna in their 2019 paper "On the Expressive power of Polynomial Neural Networks". We will look at 1) what a polynomial neural network is and how we can interpret the output such networks as varieties, 2) why the dimension of this variety and the expressive power of this network are related, and 3) how the study of these varieties might tell us something about the architecture of the network.

May 5

Ellie Thieu
Title: Visualizing Cohomology
Abstract: We will go through Ravi’s picture book together. While thinking about how to present it, I was faced with the choice of either redrawing the whole picture book, or just be honest about my cheating and use the author’s very own beautiful illustrations. This way of looking at cohomology is not perfect, but it offers a very simple understanding of taking cohomology and spectral sequences. I will deliver how to visualize cohomology, and declare it an early victory. Then we will go as far as time allow to understand spectral sequences.

Please bring your laptop, or anything you can use to follow the illustrations. Because, alas, I will not redraw them on the board.

Past Semesters

Fall 2021

Spring 2021

Fall 2020

Spring 2020

Fall 2019

Spring 2019

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

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