Graduate Algebraic Geometry Seminar Fall 2021

When: 5:00-6:00 PM Thursdays

Where: TBD

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, Colin Crowley.

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.

Fall 2021 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.

Stacks for Kindergarteners
Motives for Kindergarteners
Applications of Beilinson resolution of the diagonal, Fourier Mukai transforms in general
Wth did June Huh do and what is combinatorial hodge theory?
Computing things about Toric varieties
Reductive groups and flag varieties
Introduction to arithmetic geometry -- what are some big picture ideas of what "goes wrong" when not over an algebraically closed field?
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 (click to see abstract)
September 30	Yifan Wei	On Chow groups and K groups
October 7	Owen Goff	Roguish Noncommutativity and the Onsager Algebra
October 14	Peter YI WEI	Pathologies in Algebraic Geometry
October 21	Asvin G	Introduction to Arithmetic Schemes
October 28	Caitlyn Booms	Classifying Varieties of Minimal Degree
November 4	John Cobb	Syzygies and Koszul Cohomology
November 11	Colin Crowley	Introduction to Geometric Invariant Theory
November 18	Connor Simpson	Combinatorial Hodge Theory
December 2	Alex Mine	Fourier-Mukai Transforms
December 9	Yu Luo	Stacks for Kindergarteners

September 30

Yifan Wei

Title: On Chow groups and K groups

Abstract:

We define Chow groups and K groups for non-singular varieties, illustrate some basic properties, and explain how intersection theory is done using K groups (on a smooth surface). Then we proceed to compute the K group of a non-singular curve. On higher dimensions there might be some issues, if time permits we will show how these issues can be mitigated, and why Grothendieck-Riemann-Roch is one of the greatest theorems in algebraic geometry (in my humble opinion).

October 7

Owen Goff

Title: Roguish Noncommutativity and the Onsager Algebra

Abstract:

While throughout algebraic geometry and many other fields we like commutative rings, we often wonder what happens if our ring is not commutative. Say, for instance, you have A^2, but instead of xy=yx you have a relation xy = qyx for some constant q. In this talk I will discuss the consequences of this relation and how it relates to an object of combinatorial nature called the q-Onsager algebra.

October 14

Peter YI WEI

Title: Pathologies in Algebraic Geometry

Abstract: This talk serves as a brief discussion on pathologies in algebraic geometry, inspired by a short thread of Daniel Litt’s twitter. No hard preliminaries! :)

October 21

Asvin G

Title: Introduction to Arithmetic Schemes

Abstract: Many of us are comfortable working with varieties over the complex numbers (or other fields) but part of the magic is that it's almost as easy to consider varieties over more exotic rings like the integers or the p-adics.

I'll explain how to think about such varieties and then use them to prove the birational invariance of Hodge numbers for Calabi-Yau's over the complex numbers using results from finite fields and p-adic analysis!

TBD

October 28

Caitlyn Booms

Title: Classifying Varieties of Minimal Degree

Abstract:

The degree of a variety embedded in projective space is a well-defined invariant, and there is a sense in which some varieties have minimal degree. Long ago, Del Pezzo and Bertini classified geometrically all possible projective varieties of minimal degree. More recently, Eisenbud and Goto gave an algebraic notion that classifies such varieties. In this talk, we will introduce the necessary background and explore these two theorems and the ways they are connected.

November 4

John Cobb

Title: Syzygies and Koszul Cohomology

Abstract:

Early on in the history of algebraic geometry it was recognized that many properties/invariants of projective varieties could be deduced by looking at their hyperplane sections. Starting in the 1950s, this classical picture was gradually refined into general theory by people like Serre and Kodaira — many hard-earned numbers could now be obtained by more brainless methods. I hope to motivate a few ideas introduced in the 1980’s as a continuation of this story beginning from Serre’s vanishing theorem.

November 11

Colin Crowley

Title: Introduction to Geometric Invariant Theory

Abstract:

Given a group action on a variety, is there a quotient variety? How do you construct it? Geometric invariant theory gives partial answers to these questions for projective varieties and a particular class of groups (reductive groups). I’ll give an overview of how GIT quotients work, which will be in the language of Hartshorne chapter one and does not require any knowledge of schemes. (Although I may need to talk a little about ample line bundles. I haven't decided yet.)

With the remaining time I'll sketch how these ideas are used in constructing (coarse) moduli spaces of semistable vector bundles, and mention which areas of math use these ideas today.

November 18

Connor Simpson

Title: Combinatorial Hodge Theory

Abstract: TBD

December 2

Alex Mine

Title: Fourier-Mukai Transforms

Abstract:

I'll say a few things about derived category of sheaves and talk about Fourier-Mukai transforms, which are certain functors between the derived categories of sheaves on two schemes. In particular, I will try to elucidate what is so "Fourier" about them.