Graduate Algebraic Geometry Seminar Fall 2022

When: 1:30-2:30 PM on Fridays

Where: Van Vleck B219

Toby the OFFICIAL mascot of GAGS!!

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

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

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

Organizers: John Cobb, Yu (Joey) Luo

Give a talk!

We need volunteers to give talks this semester. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the main page. Sign up here: https://forms.gle/XUAq1VFFqqErKDEh6.

Fall 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
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
Weil Conjectures, GAGA theorems, surfaces of general type, moduli spaces, moduli of curves, mixed characteristics (stuff), elliptic curves, abelian varieties, hyperelliptic curves, resolution of singularities, minimal model program (stuff).

Being an audience member

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

Do Not Speak For/Over the Speaker
Ask Questions Appropriately

Talks

Date	Speaker	Title
September 23	Yiyu Wang	The Cox Ring of Toric Varieties
September 30	Asvin G.	The moduli space of curves
October 7	Alex Hof	Revenge of the Classical Topology
October 14	John Cobb	Virtual Resolutions and Syzygies
October 21	Yifan	TBA
October 28	Ivan Aidun	Gröbner Bases and Computations in Algebraic Geometry
November 2	Some Matroid Person	TBA
November 11	Connor Simpson	A miraculous theorem of Brion
November 18	Alex Mine	TBA
December 2	Kevin Dao	TBA
December 9	Yu Luo	TBA

September 23

Yiyu Wang

Title: The Cox Ring of Toric Varieties

Abstract: This talk will include two parts. In the first part, I will briefly introduce toric varieties, and give some examples. I will also explain how they are related to the combinatorial objects called fans. Only some basic algebraic geometry will be used in this part. In the second part, I will talk about Cox's construction of representing any toric variety as a quotient space, and his famous Cox ring. As a corollary, we can prove that the automorphism group of a complete simplicial toric variety is a linear algebraic group. I will use some basic knowledge of toric varieties in this part.

September 30

Asvin G

Title: The moduli space of curves

Abstract: I'll give a brief introduction to moduli spaces and focus mostly on the moduli spaces of genus 0 curves with marked points. These spaces are at the same time quite explicit and easy to describe while also having connections with many interesting parts of mathematics. I'll try to keep the topic fairly elementary.

October 7

Alex Hof

Title: Revenge of the Classical Topology

Abstract: The Zariski topology is pretty cool, but, if we're working over the complex numbers, we can also think about the classical topology we're used to from other areas of math. In this talk, we'll discuss analytification, the process of passing from an algebraic variety (or scheme) to the corresponding classical object (or complex-analytic space), and touch on various facts about the relationship between the two, such as Serre's GAGA principle. If time permits, we'll also talk a little about tools we can use to gain insight about varieties once we've analytified them, such as the theory of stratifications.

October 14

John Cobb

Title: Virtual Resolutions and Syzygies

Abstract: Starting from polynomials and proceeding with specific examples, the first part of this talk is dedicated to motivating the idea of syzygies and the geometric information they encode about projective varieties. This will lead us to a current focus of research: How can we use these important tools when our variety is not projective? At least in the situation of toric varieties, we can use a generalization called virtual syzygies. The second part of the talk will focus on answering a few basic questions about these analogues, such as: How can we construct examples? How complicated do they get (at least in the case of curves)?

October 21

Yifan

Title:

Abstract:

October 28

Ivan Aidun

Title: Gröbner Bases and Computations in Algebraic Geometry

Abstract: Gröbner bases are the most important computational tools in algebraic geometry and commutative algebra. They can be computed by an algorithm which simultaneously generalizes row reduction for matrices and the Eucliden algorithm for polynomial division. This algorithm has two peculiar properties: its worst-case time complexity is doubly exponential in the number of variables, but it also runs quickly on most examples of practical interest. What could account for the difference between the expected and actual runtime of this algorithm?

November 2

Some Matroid Person

Title:

Abstract:

November 11

Connor Simpson

Title: A miraculous theorem of Brion

Abstract: Schubert varieties give a basis for the cohomology ring of the grassmannian. We'll discuss a heorem of Brion, which says that subvarieties of the grassmannian whose expression in the schubert basis uses only 0 and 1 coefficients have numerous nice properties.