Graduate Algebraic Geometry Seminar Fall 2024

When: 3:30PM - 4:30PM every Wednesday starting September 4th, 2024. Talks are for 30 minutes - 1 hour with extra time for questions.

Where: Sterling 2329

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 with Andrei Căldăraru for Fall 2024.

Organizers: Kevin Dao, Yu (Joey) Luo, and Boyana Martinova.

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/Bp4bFBiS7Y6RH7mq8.

New Wishlist as of Fall 2024

This wishlist is based on requests from graduate students (new and old). Don't be intimidated by the list (especially as a new graduate student), a lot of the topics here are advanced. You are always welcome to give a talk on a topic that does not appear on this list. If you are looking for a topic and none of the ones listed below sound compelling to you, you can always reach out to one of the organizers for more ideas!

• 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.
• There are many many classes of varieties out there that people are interested -- pick one and it could very well be a talk on its own! Here are a few examples; abelian varieties, secant varieties, tangent varieties, Kazdan-Lutszig varieties, toric varieties, flag varieties, Fano varieties, Prym varieties, and beyond.

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
September 4th	GAGS SOCIAL!	No talk this week :)	But still come to hang out :) There (tentatively) will be food!
September 11th	Ivan Aidun	Who is a variety? Why is a scheme?	In this talk I will introduce the principle underlying all of algebraic geometry: the correspondence between ideals in a polynomial ring C[x_1,...,x_n] and subsets of C^n "cut out" by polynomial equations — which are called "varieties".  I will discuss several examples, ranging from familiar curves and surfaces we teach to calculus students, to more unfamiliar objects that nevertheless are very mathematically useful.  The power of algebraic geometry is that we can think of all of these objects "on a par" with one another and translate freely between geometric intuition and algebraic rigor.  Towards the end, I will discuss some of the limitations of varieties, and the things which led people to develop more general notions of what a "geometric space" could mean.
September 18th	Kevin Dao	(Non)singular Varieties with a View Towards Birational Geometry	I plan to discuss the notion of nonsingular varieties and the basics of birational geometry with an eye towards stating (obviously not proving) the resolutions of singularities of Hironaka in characteristic zero. The key point being that singularities are interesting, valuable to understand, and just because life is simpler without singularities is not enough of a justification to ignore them.
September 25th	Bella Finkel	Neurovarieties and expressiveness of polynomial neural networks	Polynomial neural networks have been implemented in a range of applications and present an advantageous framework for theoretical machine learning.The Zariski closure of the space of functions expressible by a polynomial neural network is an algebraic variety known as the network's neurovariety. We'll discuss some special cases where the neurovariety is as big as possible and translations to a combinatorial number theory problem. Based on recent work with J. Rodriguez, C. Wu, and T. Yahl.
October 2nd	Oskar Henriksson	Secant varieties with a view toward applications	Secant varieties are beautiful geometric objects that have played a central role in the history of algebraic geometry. In addition to this, they also appear naturally in many applications. In this talk, we’ll revisit some classical results about the dimension theory of secant varieties, including the celebrated Alexander–Hirschowitz theorem, and discuss a couple of examples where these results have found applications lately, ranging from rigidity theory to optimization and statistics.
October 9th	Caroline Nunn	The analogy between Galois theory and Geometry	So-called "Galois correspondences" - one to one correspondences between certain objects and subgroups of some associated group - are ubiquitous in modern mathematics. In this talk, we will look at two of the oldest examples, namely Galois theory and covering space theory. Then, we will look at how algebraic geometry shows us that the connection between the two might be more than just an analogy. No prior experience in Galois or covering space theory is necessary, but it might be helpful to have seen at least one of them.
October 16th	Ari Davidovsky	Configuration Spaces	This talk aims to show some of the ways that number theorists talk about algebraic geometry. Using configuration space as our motivating example, we will see how we can jump between algebraic geometry over $\mathbb{F}_p$ and algebraic geometry and topology over $\C$. This talk will use a lot of topology over $\C$, so it will be helpful to have seen the fundamental groups. Also, one of the goals is to talk about etale cohomology, so the later parts of the talk will also assume some familiarity with homology and cohomology at the level of 752.
SPECIAL TIME: October 21st IN VAN VLECK B119, AT 3:30pm-4:30pm.	Yiyu Wang	An Introduction to Borel–Weil–Bott Theorem	The celebrated Borel-Weil-Bott theorem which generalizes the Borel-Weil theorem, computes the cohomology of the line bundles on a flag variety on the algebra-geometric side and gives a representation of a given highest weight on the representation theoretical side. I will explain the statement by studying the line bundles on $\mathbb{P}^1$. I will also talk about the proof of the Borel-Weil theorem. If time permits, I will discuss the proof of the Borel-Weil-Bott theorem. This talk requires some knowledge on complex lie groups and their representation theory.
October 23rd	Boyana Martinova	Nonvanishing Syzygies of Veronese Embeddings	Betti tables are a well-studied structure that summarize key information from the resolution and syzygies of a module, informing our understanding of the module's complexity. However, there are extremely natural families of examples (spoiler: it's Veronese embeddings) where the Betti tables are entirely unknown, even in low-dimensions and small Veronese degrees. In this talk, I'll describe a surprisingly simple method from Ein-Erman-Lazarsfeld that identifies which Betti numbers are non-zero for Veronese embeddings, which is often enough. If time permits, I'll also briefly discuss recent progress I've made in extending this method to Veronese subrings in the non-standard graded setting.
October 30th	Caitlin Davis	An Introduction to Weighted Projective Space	When we work with nonstandard graded rings and modules, the corresponding geometric setting is not affine space or projective space, but instead weighted projective space.  While the definition of these spaces is very similar to that of usual projective space, they have some interesting geometric properties which we don't see in usual projective space.  In this talk, I'll define weighted projective space, talk about some properties (through examples), and give some motivation for why we might want to study these spaces.  In particular, these spaces show up implicitly even in classical algebraic geometry.  We'll talk about one instance of this, which has to do with hyperelliptic curves.
November 6th	Hannah Ashbach	I don't know about Grassmannians and at this point I'm afraid to ask	A gentle introduction to Grassmannians for those who are unfamiliar and perhaps afraid.
November 13th	Mohammadali Aligholi	Deligne's dictionary, from Hodge structures to l-adic structures	I'll try to explain some of Deligne's questions and ideas, presented in his 1970 ICM talk. What additional structures (on cohomology) arise when studying a variety over C, finite fields, or local fields, and how are they related? Deligne suggested that there are striking analogies. These vague questions led to many deep conjectures and motivated him to develop the full theory of mixed Hodge structures within two years and to prove the Riemann hypothesis (over finite fields) four years later.
SPECIAL TIME: November 14th IN BIRGE 348 AT 2:00pm-3:00pm.	Yunfeng Jiang	Enumerative geometry for KSBA spaces	Pre-Talk for AG Seminar Talk
November 20th	Owen Goff	Spiced Arithmetic Geometry	Today we will bring together many of the topics we've mentioned this semester into the diverse field of arithmetic geometry and see how algebraic geometry works over fields and rings over than C.
November 27th	NO GAGS - THANKSGIVING	ENJOY YOUR HOLIDAY OFF	ENJOY YOUR HOLIDAY OFF
December 4th	Ruocheng Yang	Stories Around the Moduli of Curves	I will discuss how Riemann originally computed that the dimension of the moduli space of curves is 3g-3 and explore how this space can be compactified. If time permits, I will introduce how to perform intersection theory on the compactified moduli space. The talk will include plenty of examples to illustrate these concepts!
December 11th	Alex Menzia	Topological Field Theories and Frobenius Algebras	I plan to first define Topological Field Theories and Frobenius Algebras, then sketch a proof of why they are equivalent, and finally with whatever remaining time I have explore various generalizations of each concept (and the extent to which the equivalence still holds).

Past Semesters

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