Graduate Algebraic Geometry Seminar Spring 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

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
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
April 21	Connor Simpson
April 28	Karan	Using varieties to study polynomial neural networks
May 5	Ellie Thieu

February 10

Everyone

Title: Informal chat session

Abstract: Bring your questions!

February 17

Asvin G

Title: TBD

Abstract: TBD

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. \end{abstract}

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: Chow rings for polymatroids via fans

Abstract: Matroids abstract the combinatorics of hyperplane arrangements. They are generalized by polymatroids, which abstract the combinatorics of subspace arrangements. Many problems on matroids have recently succumbed to algebraic geometry-inspired methods, often involving the "Chow ring" of a matroid. We will discuss Pagaria & Pezzoli's generalization of this ring to polymatroids. We show the Chow ring of a polymatroid can be realized as the Chow ring of a smooth fan. Applications include a new proof of Hodge theory for the Chow ring of a polymatroid. This is joint work with Colin Crowley, June Huh, Matt Larson, and Botong Wang.

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:

Abstract:

Past Semesters

Fall 2021

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Fall 2019

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Fall 2015