Graduate Algebraic Geometry Seminar Spring 2024

When: 2:30PM - 4:00PM every Wednesday starting January 31st, 2024. Talks are for 30 minutes - 1 hour with extra time for questions.

Where: Van Vleck B325

Toby the OFFICIAL mascot of GAGS!!

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

Why: The purpose of this seminar is to learn algebraic geometry, commutative algebra, and broadly algebra itself, by giving and listening to talks in a informal setting. Sometimes people present an interesting topic or 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, 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.

Enrollment in Math 941: The correct section to enroll for Math 941 is is with primary instructor being Dima Arinkin. If you are signed up for this section, you are expected to give a talk to get a grade.

Organizers: John Cobb, Kevin Dao, Yu (Joey) Luo.

Feedback Form for Organizers: The form is anonymous. You can find it here.

Give a talk!

We need volunteers to give talks. Beginning graduate students are encouraged to give a talk. If you need ideas, see the wish list below, the main page, or talk to an organizer. It is expected that people enrolled in Math 941: Seminar in Algebra must give a talk to get credit. Sign up sheet: https://forms.gle/JofcgHVZyQmEKpcX7.

New Wishlist as of Spring 2024

The following is a list of topics that would be good to have in a graduate algebra and algebraic geometry seminar.

• Topics in Representation Theory. There are many topics one can discussion: explaining Lie algebra representations via Fulton-Harris's book (Lecture 7-9), Brauer theory, the Stone-von Neumann theorem, classification and determination of unitary representations, the Harish-Chandra isomorphism, Borel-Bott-Weil, historical results such as Frobenius determinants. Quiver representations are another topic; there is a well-written book by Ralf Schiffler you could look at for this topic.
• The Riemann-Roch Theorem, its generalizations: Grothendieck-Riemann-Roch, Hirzebruch-Riemann-Roch, and applications.
• GAGA Theorems and how to use them. Some ideas on important results to talk about can be found here. For some references to look at: the appendix in Hartshorne's Algebraic Geometry, Serre's original GAGA paper, and Neeman's book Algebraic and Analytic Geometry.
• Cohen-Macaulay rings and schemes and variants of this type. A useful topic for those working with "mild singularities". The standard reference for this stuff is the book by Brunz and Herzog, but Eisenbud's Commutative Algebra book also has a lot of things to say about CM rings.
• Hodge Theory for the working Algebraic Geometer. What is the Hodge decomposition? What is the Hard Lefschetz Theorem? What is the statement of the Hodge conjecture? Dolbeault cohomology?
• Algebraic Curves via Hartshorne Chapter IV. What can be said projective curves of degree d and genus g? How do (did) people study algebraic curves? What are the important facts about curves a working algebraic geometer should know?
• Algebraic Suraces via Hartshorne Chapter V and Beauville's Complex Algebraic Surfaces. What does the birational classification of complex algebraic surfaces look like? How should we classify objects?
• Vector Bundles on P^n. A good reference for this is "Vector Bundles on Complex Projective Spaces" by Christian Okonek. Interesting points of discussion could inclued any of: Horrock's Criterion for vector bundles, Beilinson's Theorem, splitting of uniform bundles of rank r<n, moduli of stable 2-bundles, constructions of vector bundles on P^n for low values of n, Serre's construction of rank 2 bundles, proof of the Grothendieck-Birkhoff Theorem, and etc. These are all very classical problems / theorems in algebraic geometry and a talk on these topics would make a great expository talk.
• Basics of Moduli: functor of points, representable functors, moduli of curves M_g, moduli of Abelian varieties of dimension g, and why do we care? A reference is Harris and Morrison, but there is the now growing textbook by Jarod Alper titled "Stacks and Moduli". Lots of lots of examples are encouraged.
• What is a syzygy? Compute some minimal free resolutions and tell people about how syzygies can tell you a lot about a curve. The Geometry of Syzygies by David Eisenbud is also a good reference and introduction to this topic.
• Derived categories and the Fourier-Mukai Transform. Introduce derived categories and explain their importance in algebraic geometry e.g. through the Fourier-Mukai transform. The book "Fourier-Mukai Transforms in Algebraic Geometry" by Daniel Huybrechts is a good reference for this stuff, but there is also the notes by Andrei Căldăraru on the Arxiv which are more to the point.
• Introduction to Algebraic Stacks: there are a number of references for this e.g. Alper's notes on Moduli, "Algebraic Stacks" by Tomas L. Gomez, the original paper of Deligne and Mumford titled "The Irreducibility of the Space of Curves of Given Genus", Martin Olsson's book "Algebraic Spaces and Stacks", and so on. Examples would be strongly encouraged over technical details and Alper's notes and/or Gomez's article are the best for this.

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 abstract algebra, algebraic geometry, representation theory, 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
• Save lengthy questions or highly technical questions for after the talk

Talks

Date	Speaker	Title	Abstract
01-31-2024	Kevin Dao	Setting up GAGS + A Survival Guide to Sheaf Cohomology.	Discussion about GAGS expectations + getting list of speakers. The short talk shall be about the basics of sheaf cohomology and all about telling the audience what they need to start computing things. The goal is to prove the genus-degree formula for smooth curves in the projective plane.
02-07-2024	Boyana Martinova	An Introduction to Cohen-Macaulay Rings	In this talk, I will introduce Cohen-Macaulay rings and discuss some key properties that make them a desirable class of rings to study. We'll explore some key techniques for determining whether a ring is Cohen-Macaulay and test various examples along the way. I plan to build all the dimension theory that is needed as we go, so this talk should be especially accessible to early graduate students.
02-14-2024	Caitlin Davis	Introduction to the Rational Normal Curve	The rational normal curve is an important example of many nice algebraic and geometric properties. I will discuss some of these properties, focusing on small concrete examples. This talk will aim to be accessible to grad students who have taken a semester or two of abstract algebra, and will not assume much (if any) algebraic geometry background.
02-21-2024	Jack Messina	Introducing Nonabelian Hodge Theory	This talk introduces the Nonabelian Hodge correspondence, which for a smooth complex projective variety relates vector bundles on it with a flat connection to certain bundles called Higgs bundles on the variety. We will introduce the relevant objects, discuss the proof of the nonabelian Hodge correspondence, and describe some interpretations of the correspondence, in particular as giving a Hodge decomposition for cohomology with nonabelian coefficients.
02-28-2024	Yiyu Wang	An Introduction to the Schubert Calculus	Some enumerative problems can be interpreted as counting the intersection points of two subvarieties of the Grassmannian, an idea that dates back to the nineteenth century. Here is a typical question: How many lines in 3-space, in general, intersect four given lines? In this talk, we will talk about how this is solved in the framework of algebraic geometry.
03-06-2024	Alex Mine	Gorenstein Rings and Duality	In this talk, I will discuss duality in algebraic geometry, emphasizing the role of Gorenstein rings. I'll demonstrate how these ideas relate to singular curves and chicken McNuggets. My goal will be to keep (almost) everything very down to earth and accessible.
03-13-2024	Bella Finkel Holman	Introduction to Tropical Geometry	Tropical geometry is a relatively new field lying at the interface of combinatorics and algebraic geometry with broad applications; tropical mathematics has been used to answer questions about the arithmetic of abelian varieties and integrable systems and applied in approaches to mirror symmetry, to name a few examples. We will introduce the objects of tropical geometry and describe the connection between classical and tropical varieties, focusing on analogs of classical results in the tropical context.
03-20-2024	Jacob Wood	Grassmannians	I will talk about Grassmannians mostly from the topological point of view. Touch some Grass-mannians.
04-03-2024	Ruocheng Yang	Ample, Nef, and Big Line Bundles	This presentation introduces ample and nef line bundles in algebraic geometry, focusing on concrete examples and classical definitions. We illustrate how these bundles are crucial for understanding projective varieties and moduli spaces, emphasizing their role in the early stages of birational geometry.
04-10-2024	Kaiyi Huang	Polynomial methods for the endpoint multilinear Kakeya problem	I will sketch the proof of Larry Guth’s celebrating result in the endpoint multilinear Kakeya problem (intersection theory in a different flavour). The main tools are Borsuk-Ulam Theorem and properties of low degree polynomials.
04-17-2024	Ivan Aidun	Dreaming AG in Technicolor	The full power of AG depends on our ability to translate our visual intuition about pictures into algebra and visa versa. In this talk, I'll talk about some of the translations that people are often forced to "pick up along the way". I'm flexible in what I talk about, depending on interest, but some possibilities (in roughly ascending order of nicheness) are: complex points; projective varieties and their affine cones (what do Erman's students know that we don't??); "fuzz"; localizations, generic points, and DVRs; the spectra of C[x], R[x], and Z[x]; line bundles and invertible sheaves; sheaves more generally; flat maps (and normal cones??); formal neighborhoods; (filthy filthy filthy filthy) arithmetic behavior; elliptic curves in various views; étale maps.
04-24-2024	Jameson Auger	Calculation of Local Fourier Transforms	For any D-module on k[z,\partial], we can describe its Fourier transform by interchanging the actions of z and \partial. We can also look at this locally using connections on vector spaces over the field of formal Laurent series. We will explicitly compute the result of this local Fourier transform using the classification of vector spaces with connections.
05-01-2024	Gabriela Brown	Group Determinants and the Origin of Representation Theory	When Dedekind and Frobenius first discovered representation theory, they were trying to factor something called the group determinant. The goal of this talk is to introduce the group determinant while describing the questions that motivated representation theory. This historical journey will include some Galois theory, lots of character theory, and end with the modern representation theory perspective on the group determinant.

Past Semesters

Fall 2023 Spring 2023

Fall 2022 Spring 2022

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