Difference between revisions of "Graduate Algebraic Geometry Seminar"
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 bgcolor="#A6B658" align="center" style="fontsize:125%"  '''Erika Pirnes'''   bgcolor="#A6B658" align="center" style="fontsize:125%"  '''Erika Pirnes'''  
    
−   bgcolor="#BCD2EE" align="center"  Title:  +   bgcolor="#BCD2EE" align="center"  Title: BuchsbaumEisenbudHorrocks conjecture 
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−  Abstract:  +  Abstract: Betti numbers are defined to be the ranks of the free modules in the free resolution of a module. The BuchsbaumEisenbudHorrocks conjecture gives upper bounds for the Betti numbers. I'll state the conjecture and give some examples. 
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Revision as of 18:28, 5 December 2019
When: Wednesdays 4:25pm
Where: Van Vleck B317
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.)
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 AxGrothendieck 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 MadsenWeiss 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 selfintersection 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 Mcurves of DarmonGranville, 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 AxGrothendieck theorem and other fun stuff 
Abstract: The AxGrothendieck 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 algebrogeometric 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 
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 Smodules), 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: BuchsbaumEisenbudHorrocks conjecture

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