Graduate Algebraic Geometry Seminar Spring 2022
When: 4:30-5:30 PM Thursdays
Where: VV B231
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: email@example.com by sending an email to firstname.lastname@example.org. 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
|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|
|Title: Informal chat session|
|Abstract: Bring your questions!|
|Abstract: Some motivation behind motives|
|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.
|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.|
|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.|
|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.|
|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.|
|Title: Symplectic geometry and invariant theory|
|Abstract: We discuss connections between symplectic geometry and invariant theory.|
|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.|
|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.