Graduate Algebraic Geometry Seminar Fall 2017
When: Wednesdays 4:40pm
Where:Van Vleck B321 (Spring 2017)
Why: The purpose of this seminar is to learn algebraic geometry 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.
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: firstname.lastname@example.org. The list registration page is here.
Give a talk!
We need volunteers to give talks this semester. If you're interested contact DJ, 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.
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)
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.)
|Date||Speaker||Title (click to see abstract)|
|January 25||Nathan Clement||Hodge to de Rham, part one|
|February 1||Nathan Clement||Hodge to de Rham, part two|
|February 8||Dima Arinkin||Motivated introduction to geometric Langlands|
|February 15||No Talk||We Failed, We All Failed|
|February 22||No Talk||We Failed, We All Failed Pt. 2|
|March 1||Brandon Boggess||An Introduction to Mori's Program|
|March 8||David Wagner||Picard groups of moduli problems|
|March 15||No Talk||We Failed, We All Failed Pt. 3|
|March 22||Spring Break||No Seminar.|
|March 29||David Wagner||Picard groups of moduli problems II|
|April 5||John Wiltshire-Gordon||Adjoint functors rule your life|
|April 12||Wendy Cheng||Boij-Soderberg Theory|
|Title: Hodge to de Rham, part one|
Abstract: I will use the magic of differential calculus in positive characteristic to prove an important result in the cohomology of smooth varieties in positive characteristic. The techniques I'll use are mainly elementary, but prior experience with differential forms, the Frobenius homomorphism, and a little homological algebra will help. This is the setup, come back next week for the punchline!
|Title: Hodge to de Rham, part two|
Abstract: Having proved an important result in positive characteristic, I'll give a nifty argument to leverage the positive characteristic statement into a characteristic zero result. I'll talk about some cohomology comparison theorems, and we'll see that all this business in positive characteristic provides an alternate proof to the classic Hodge decomposition theorem for cohomology.
|Title: Motivated introduction to geometric Langlands|
Abstract: The Langlands program originated from bold conjectures formulated by Robert Langlands in the late 1960's. The conjectures combine number theory and representation theory in a highly unexpected way. The geometric Langlands program adds algebro- geometric methods (and, sometimes, physics) to the mix.
This interplay of ideas creates a beautiful picture... and a very challenging subject. In my talk, I will suggest one possible way to approach the area. I plan to focus on questions rather than answers: the goal is to explain how (some of) the questions fit together, and to tell you the keywords that go into the answers.
|Title: We Failed, We All Failed Pt. 1|
|Title: We Failed, We All Failed Pt. 2|
|Title: An Introduction to Mori's Program|
Abstract: In studying the birational classification of varieties, one plan of attack is to construct a "simplest" variety in each birational equivalence class. We will see how this approach gives a full structure theory for surfaces, and investigate what new challenges arise in the case of threefolds.
|Title: Picard groups of moduli problems|
Abstract: In a pastoral traipse, I will discuss some lower bounds on the time it takes an average adult to drink a medium Frosty from Wendy's, including some results of my own about making these bounds sharp. Time permitting, I will also explain how this theory can be extended to study the ingestion of M&M's, a connection previously unkown.
|Title: We Failed, We All Failed Pt. 3|
|Title: No Seminar.|
|Title: Picard groups of moduli problems II|
Abstract: Having discussed Grothendieck topologies and an existence theorem for absolute products of families, we construct and give a convenient characterization of the line bundles on the moduli problem, finally proving that . Our numinous and mystical journey towards this sublime result will make ample use of Grothendieck's generalized Hilbert 90. Facts you knew about sheaf cohomology of schemes will suddenly materialize in the context of sites.
|Title: Adjoint functors rule your life|
Abstract: This talk is about adjoint functors. We will do examples!
|Title: Boij-Soderberg Theory|