Graduate Algebraic Geometry Seminar

When: Wednesdays 4:25pm

Where: Van Vleck B317

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. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday 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@lists.wisc.edu. The list registration page is here.

Give a talk!

We need volunteers to give talks this semester. If you're interested contact Caitlyn or David, or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.

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:

The List of Topics that we Made February 2018

On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:

Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.

Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. What is the Grassmanian, you say? That's probably a talk we should have every year, so you should give it!

Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.

Katz and Mazur explanation of what a modular form is. What is it?

Kindergarten moduli of curves.

What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?

Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)

Hodge theory for babies

What is a Néron model?

What is a crystal? What does it have to do with D-modules? Here's an encouragingly short set of notes on it.

What and why is a dessin d'enfants?

DG Schemes.

Ed Dewey's Wish List Of Olde

Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.

Here are the topics we're DYING to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.

Specifically Vague Topics

D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.

Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)

Famous Theorems

Interesting Papers & Books

Symplectic structure of the moduli space of sheaves on an abelian or K3 surface - Shigeru Mukai.

Residues and Duality - Robin Hatshorne.
- Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)

Coherent sheaves on P^n and problems in linear algebra - A. A. Beilinson.
- In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)

Frobenius splitting and cohomology vanishing for Schubert varieties - V.B. Mehta and A. Ramanathan.
- In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off!

Schubert Calculus - S. L. Kleiman and Dan Laksov.
- An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!

Rational Isogenies of Prime Degree - Barry Mazur.
- In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.

Esquisse d’une programme - Alexander Grothendieck.
- Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)

Géométrie algébraique et géométrie analytique - J.P. Serre.
- A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)

Limit linear series: Basic theory- David Eisenbud and Joe Harris.
- One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.

Picard Groups of Moduli Problems - David Mumford.
- This paper is essentially the origin of algebraic stacks.

The Structure of Algebraic Threefolds: An Introduction to Mori's Program - Janos Kollar
- This paper is an introduction to Mori's famous ``minimal model program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties.

Cayley-Bacharach Formulas - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.
- A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?.

On Varieties of Minimal Degree (A Centennial Approach) - David Eisenbud and Joe Harris.
- Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.

The Gromov-Witten potential associated to a TCFT - Kevin J. Costello.
- This seems incredibly interesting, but fairing warning this paper has been described as highly technical, which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)

Fall 2019

Date	Speaker	Title (click to see abstract)
September 18	David Wagner	M_g Potpourri
September 25	Shengyuan Huang	Derived Groups and Groupoids
October 9	Brandon Boggess	Geometry of Generalized Fermat Curves
October 16	Soumya Sankar	Brauer groups and obstruction problems
October 23	Alex Mine	The Ax-Grothendieck theorem and other fun stuff
October 30	Vlad Sotirov	Buildings and algebraic groups
November 6	Connor Simpson	Lorentzian Polynomials
November 13	Alex Hof	Tropicalization Blues
November 20	Caitlyn Booms	Computing Gröbner Bases of Submodules
November 27	Thanksgiving Break
December 4	Colin Crowley	Title TBD
December 11	Erika Pirnes	Title TBD

September 18

David Wagner

Title: M_g Potpourri

Abstract: In 1983, David Mumford proposed that the rational cohomology ring of Mg should be a polynomial algebra. I will discuss some of the history of Mumford's conjecture, possibly indicating a few ideas from the 2007 proof as the Madsen-Weiss theorem. If all goes well, the talk will take us through such diverse places as homotopy theory, representation stability, combinatorics of ribbon graph complexes, and deformations of algebras.

September 25

Shengyuan Huang

Title: Derived Groups and Groupoids

Abstract: In this talk, we will discuss groups and groupoids in the derived category of dg schemes. I will focus on examples instead of the abstract theory. If X is a smooth subscheme of a smooth scheme S over the field of complex numbers, then the derived self-intersection of X in S is a groupoid. We will investigate the corresponding Lie algebroid of the groupoid I mentioned above, and exponential map between them.

October 9

Brandon Boggess

Title: Geometry of Generalized Fermat Curves

Abstract: We will look at the generalized Fermat equation, and see how studying its integral points leads one to study quotient stacks. We will then very quickly turn and run away from the general picture to a particularly simple example of these quotient stacks, the M-curves of Darmon-Granville, and how they can be used to say something about integral points without having to actually know what the hell a stack is.

October 16

Soumya Sankar

Title: Brauer groups and obstruction problems

Abstract: Brauer groups are ubiquitous in arithmetic and algebraic geometry. I will try to describe different contexts in which they appear, ranging from Brauer groups of fields and class field theory, to obstructions to moduli problems and derived equivalences.

October 23

Alex Mine

Title: The Ax-Grothendieck theorem and other fun stuff

Abstract: The Ax-Grothendieck theorem says that any polynomial map from C^n to C^n that is injective is also surjective. The way this is proven is to note that the statement is trivial over finite fields, and somehow use this to work up to the complex numbers. We'll talk about this and other ways of translating information between finite fields and C.

October 30

Vlad Sotirov

Title: Buildings and algebraic groups

Abstract: I will give a concrete introduction to the notion of a Tits building and its relationship to algebraic groups.

November 6

Connor Simpson

Title: Lorentzian Polynomials

Abstract: Lorentzian polynomials are a family of multivariate polynomials recently introduced by Branden and Huh. We will define Lorentzian polynomials and survey some of their applications to combinatorics, representation theory, and computer science. The first 20 minutes of this talk should not require more than the ability to take partial derivatives of polynomials and basic linear algebra.

November 13

Alex Hof

Title: Tropicalization Blues

Abstract: Tropicalization turns algebro-geometric objects into piecewise linear ones which can then be studied through the lens of combinatorics. In this talk, I will introduce the basic construction, then discuss some of the recent efforts to generalize and improve upon it, touching upon the Giansiracusa tropicalization and ~~developing~~ gazing wistfully in the direction of the machinery of ordered blueprints necessary for the Lorscheid tropicalization.

November 20

Caitlyn Booms

Title: Computing Gröbner Bases of Submodules

Abstract: In this talk, we will give motivation for and define Gröbner bases of submodules of finitely generated free modules over a polynomial ring S=k[x_1,...,x_r]. Not only are such bases extremely useful in constructive module theory and elimination theory, they are actually computable thanks to Buchberger's Algorithm. Further, they have a wide variety of applications in algebraic geometry including aiding in the computation of syzygies (kernels of maps of finitely generated, free S-modules), Hilbert functions, intersections of submodules, saturations, annihilators, projective closures, and elimination ideals. We will work through several examples and discuss some of these applications.

November 28

Thanksgiving Break

Title:

Abstract:

December 4

Colin Crowley

Title:

Abstract:

December 11

Erika Pirnes

Title: Buchsbaum-Eisenbud-Horrocks conjecture

Abstract: Betti numbers are defined to be the ranks of the free modules in the free resolution of a module. The Buchsbaum-Eisenbud-Horrocks conjecture gives upper bounds for the Betti numbers. I'll state the conjecture and give some examples.