Difference between revisions of "Graduate Algebraic Geometry Seminar Fall 2017"
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Revision as of 18:02, 14 November 2017
When: Wednesdays 3:30pm
Where:Van Vleck B321 (Fall 2017)
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 Juliette, 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:
Wish List
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 2017
Date  Speaker  Title (click to see abstract) 
September 13  Moisés Herradón Cueto  Vector bundles over the projective line 
September 20  No Talk  Reflecting on signing up for a talk 
September 27  Moisés Herradón Cueto  Vector bundles over an elliptic curve 
October 4  TBD  TBD 
October 11  TBD  TBD 
October 18  TBD  TBD 
October 25  TBD  TBD 
November 1  Michael Brown  A Theorem of Orlov 
November 8  Michael Brown  A Theorem or Orlov 
November 15  TBD  TBD 
November 22  n/a  No Seminar 
November 29  TBD  TBD 
December 6  TBD  What about stacks? 
December 13  TBD  What about stacks? II 
September 13
Moisés Herradón Cueto 
Title: Vector Bundles over the projective line 
Abstract: Next week I will do an overview of Atiyah's classification of bundles on an elliptic curve. Today, I will talk about the tools needed to do this: cohomology of vector bundles. My goal is to keep a loose, islander, Ibizan pace where I will not define anything very rigorously, yet we will get our hands dirty with some computations, not all of which you have sat down and done before (if you have, what is your life? Why am I the one giving this talk?). Our aimless drift will hopefully get us to the much easier classification of vector bundles on the projective line, and we will have achieved the feat of using cohomology to prove a statement that doesn't contain the word cohomology! Flowery crowns are optional. 
September 20
No talk 
Title: You should sign up to give a talk 
Abstract: TBD 
September 27
Moisés Herradón Cueto 
Title: Vector bundles over an elliptic curve 
Abstract: We will regain our continental composture and discuss Atiyah's classification of bundles on an elliptic curve. There will be a ton of preliminary stuff, some lemmas, some theorems and some sketchy proofs. The sun will rise on the east and set on the west, and in the mean time we will learn all the isomorphism classes of vector bundles on an elliptic curve over any field. 
October 4
TBD 
Title: TBD 
Abstract: TBD 
October 11
TBD 
Title: TBD 
Abstract: TBD 
October 18
TBD 
Title: TBD 
Abstract: TBD 
October 25
TBD 
Title: TBD 
Abstract: TBD 
November 1
Michael Brown 
Title: A theorem of Orlov 
Abstract: I will discuss the main theorem of Orlov's "Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities". This very powerful theorem provides a comparison between the derived category of coherent sheaves on certain schemes and a related gadget called the "singularity category". Orlov's theorem recovers Beilinson's semiorthogonal decomposition of the bounded derived category of projective space as a special case.

November 8
Michael Brown 
Title: A Theorem of Orlov 
Abstract: This will be a continuation of the previous talk. 
November 15
TBD 
Title: TBD 
Abstract: TBD 
November 22
No Seminar This Week 
Title: Enjoy Thanksgiving! 
Abstract: n/a 
November 29
TBD 
Title: TBD 
Abstract: TBD 
December 6
Nathan Clement 
Title: What about stacks? 
Abstract: TBD 
December 13
Nathan Clement 
Title: What about stacks? II 
Abstract: TBD 