Graduate Algebraic Geometry Seminar Spring 2019
When: Wednesdays 4:25pm
Where: Van Vleck B317 (Spring 2019)
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 indepth. 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 threedimensional 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 Dmodules? 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
 Dmodules 101: basics of Dmodules, equivalence between left and right Dmodules, pullbacks, pushforwards, maybe the GaussManin Connection. Claude Sabbah's introduction to the subject could be a good place to start.
 Sheaf operations on Dmodules (the point is that then you can get a FourierMukai transform between certain Omodules and certain Dmodules, 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 semiorthogonal 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 (Fregularity, strong Fregularity, Fpurity, 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, BrillNoether 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.
 CayleyBacharach Formulas  Qingchun Ren, Jürgen RichterGebert, 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 GromovWitten 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 Ainfinity algebras and the derived category of a CalabiYau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)
Spring 2019
Date  Speaker  Title (click to see abstract) 
February 6  Vlad Sotirov  Heisenberg Groups and the Fourier Transform 
February 13  David Wagner  DG potpourri 
February 20  Caitlyn Booms  Completions of Noncatenary Local Domains and UFDs 
February 27  Sun Woo Park  Baker’s Theorem 
March 6  Connor Simpson  Mason's Conjectures and Chow Rings of Matroids 
March 13  Brandon Boggess  Dial M_1,1 for moduli 
March 27  Solly Parenti  Quadratic Forms 
April 3  Colin Crowley  RiemannRoch and AbelJacobi theory on a finite graph 
April 10  Alex Hof  Kindergarten GAGA 
April 17  Soumya Sankar  Inseparable maps and quotients of varieties 
April 24  Wendy Cheng  Introduction to BoijSöderberg Theory 
May 1  Shengyuan Huang  Orbifold Singular Cohomology 
February 6
Vladimir Sotirov 
Title: Heisenberg Groups and the Fourier Transform 
Abstract: I will discuss the relationship between Fourier transforms and the Heisenberg groups, with a view toward the discussion of line bundles on complex tori that appears in Polishchuk's book Abelian Varieties, Theta functions, and the Fourier transform. 
February 13
David Wagner 
Title: DG potpourri 
Abstract: I will explain how differential graded categories made their way into AG as a way to solve some of the inadequacies of the ordinary derived category. We will then give examples of the utility of DG techniques.

February 20
Caitlyn Booms 
Title: Completions of Noncatenary Local Domains and UFDs 
Abstract: A ring is called catenary if for any pair of prime ideals P contained in Q, all saturated chains of prime ideals between P and Q have the same length.
In this talk, I will introduce the necessary background about noncatenary rings and completions of local (Noetherian) domains, as well as the relevant history. Then, I will give the characterization of completions of noncatenary local domains and noncatenary local UFDs, which I will use to describe examples of very strange rings. 
February 27
Sun Woo Park 
Title: Baker's Theorem 
Abstract: As a follow up talk to last semester, I will state and prove Baker’s theorem, a classical theorem which uses Newton Polygons to count the genus of a given curve. I will also briefly sketch how one can use Newton polygons to understand resolution of singularities of a given curve over valuation rings.
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March 6
Connor Simpson 
Title: Mason's Conjectures and Chow Rings of Matroids 
Abstract: A matroid is a combinatorial structure that abstracts many different notions of independence, including linear independence in a vector space. Mason's Conjectures are a series of three increasingly strong inequalities on certain numerical invariants of matroids, the weakest of which resisted proof for over 40 years until its resolution in 2015. However, in the years since, all of Mason's conjectures have become theorems!
In this talk, we will introduce matroids and the Chow ring of a matroid, the amazing algebraic gadget used to prove Mason's first conjecture. Finally, outline the proof of Mason's second conjecture (work of our very own Botong Wang & coauthors). 
March 13
Brandon Boggess 
Title: Dial M_1,1 for moduli 
Abstract: We will speculate wildly about what kind of object the moduli space of elliptic curves should be. I don't know what a stack is, and I promise not to try to define one.
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March 27
Solly Parenti 
Title: Quadratic Forms 
Abstract: Starting from the classical story of binary quadratic forms, we'll move on to more modern aspects of the theory of quadratic forms and try to make sense of some mass formulas. 
April 3
Colin Crowley 
Title: RiemannRoch and AbelJacobi theory on a finite graph 
Abstract: A finite graph is like a Riemann surface, or a smooth projective curve. Following the paper by Baker and Norine, we will investigate linear equivalence of divisors on graphs, the Jacobian of a graph, and a combinatorial interpretation of these as a chip firing game. 
April 10
Alex Hof 
Title: Kindergarten GAGA 
Abstract: Join me in regressing to an infantile state as we discuss Serre's 1956 paper Algebraic geometry and analytic geometry, widely considered to be the most influential work ever authored by a mathematician under the age of five. We will define the notion of an analytic space, construct the analytic space associated to any algebraic variety over [math]\displaystyle{ \mathbb{C} }[/math], and examine the relationships between the two, including the equivalence between coherent algebraic sheaves and coherent analytic sheaves in the projective case.
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April 17
Soumya Sankar 
Title: Inseparable maps and quotients of varieties 
Abstract: The theory of inseparable maps is inseparable from the study of varieties in positive characteristic, as are quotients of varieties by wonderfully nonreduced group schemes. I will talk about the theory of derivations and Lie algebras and how these are helpful in understanding both the structure of inseparable maps, as well as groupscheme actions on varieties.
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April 24
Wendy Cheng 
Title: Introduction to BoijSöderberg Theory 
Abstract: BoijSöderberg theory describes the Betti diagrams of graded modules over the polynomial ring, up to multiplication by a rational number. Analog EisenbudSchreyer theory describes the cohomology tables of vector bundles on projective spaces up to rational multiple. We give an introduction and survey of these newly developed areas. 
May 1
Shengyuan Huang 
Title: Orbifold Singular Cohomology 
Abstract: Let [X/G] be a stack which is a global quotient of a manifold X by a finite group G. There is a way to construct an orbifold singular cohomology ring. It is the correct generalization of singular cohomology of a topological space, because it coincides with the singular cohomology of a crepant resolution of the quotient space X/G. I will compute several example to explain this. Moreover, (orbifold) singular cohomology ring of a space should corresponds to the (orbifold) Hochschild cohomolgy ring of its mirror if you believe Homological Mirror Symmetry. I will briefly compare these two sides of Homological Mirror Symmetry by computing concrete examples. 