Graduate Algebraic Geometry Seminar Spring 2018
When: Wednesdays 3:40pm
Where:Van Vleck B321 (Spring 2018)
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: email@example.com. The list registration page is here.
Give a talk!
We need volunteers to give talks this semester. If you're interested contact Juliette or Moisés, 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.
- 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)
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)|
|February 14||Moisés Herradón Cueto||Fun with commutative groups|
|February 21||Moisés Herradón Cueto||[math]\mathcal F[/math]un with commutative groups|
|February 28||Dima Arinkin||ODEs: algebraic vs analytic vs formal|
|March 7||Vladimir Sotirov||Topology of Affine Schemes|
|March 14||David Wagner||Derived Quot Schemes|
|March 21||Mao Li||D modules in prime characteristic|
|March 28||Spring break||Whoo!|
|April 4||Rachel Davis||Introduction to Grothendieck's dessins d'enfants|
|April 11||Brandon Boggess||Néron models of elliptic curves|
|April 18||Soumya Sankar||Modular forms over arbitrary rings|
|April 25||Solly Parenti||The Theory of Heights and the Height of Theories|
|May 2||No talk today||Keith Rush is giving a talk at the same time|
|Moisés Herradón Cueto|
|Title: Fun with commutative groups|
My goal is to next week talk about Gerard Laumon's preprint Transformation de Fourier généralisée, which will encapsulate all the Fourier transforms I can name (which for me includes the Fourier transform for D-modules on affine space, the Mellin transform, the Fourier-Mukai transform for abelian varieties and the rare John Mahoney transform between modules on the punctured line and vector spaces with a Z-action) into one crazy package.
In order to achieve this altered mental state, we will have to rethink all our preconceptions, and rediscover algebraic groups, make sense of Cartier duality for them, define formal groups along the way, and see how duality of Abelian varieties relates to Cartier duality. If you come to the talk in an already in this altered mental state, then hopefully I'll do some examples that you've been too lazy to work out for yourself.
|[math]\mathcal F[/math]un with commutative groups|
|Title: [math]\mathcal F[/math]un with commutative groups|
I will talk about how last week's Cartier duality relates to duality for abelian varieties. Then we will see how this allows us to take certain complexes of groups which are self dual, going back to altered mental states which are now derived.
If we're going to Fourier transform something, it should be a sheaf, but what is even a sheaf on a complex of groups? Once we see what it should be, I will handwave what the Fourier transform is and give a bunch of concrete examples.
|Title: ODEs: algebraic vs analytic vs formal|
Linear ordinary differential equations (or, as they are known geometrically, bundles with connections on Riemann surfaces) can be studied from many directions. In algebraic geometry, the goal is to classify equations rather than to solve them. The classification can be done in different settings: algebraic, analytic, or formal; each setting has its advantages and disadvantages. However, for one of the most important class of equations (equations with regular singularities aka Fuchsian equations) the three approaches agree, leading to a rich and beautiful picture.
In my talk, I will sketch the algebraic theory of differential equations, focusing on equations with regular singularities. (I do not expect to have time for irregular singularities.) The talk is related to my topics course on D-modules, but it does not rely on the topics course.
|Title: Topology of Affine Schemes|
I will explain in what sense flat morphisms of (affine) schemes are the correct analogue of open maps of topological spaces, and then use that to explain how surjectivity in the sense of Zariski spectra corresponds to surjectivity of functors of points.
|Title: Derived Quot Schemes|
The titular paper of Ciocan-Fontanine & Kapranov constructs a "derived" version of Grothendieck's Quot scheme, whose degree zero truncation is the ordinary Quot scheme, and whose tangent space at each k-point consists higher Ext's. In this talk, we will talk about what is meant by "derived" in modern AG and why you might want a derived Quot scheme. In the remaining time, we will give a (hopefully transparent) description of how CF&K construct RQuot.
|Title: D-modules in prime characteristic|
The usual theory of D-modules concerns itself with sheaves and varieties over complex numbers. In this talk I'm going to consider its characteristic p analogue. The new feature is that the ring of differential operators has a large center. By using this extra information we can give a more precise description about the ring of differential operators, namely it has a canonical structure of an Azumaya algebra over the twisted cotangent bundle. This has lots of applications in algebraic geometry and representation theory.
|Introduction to Grothendieck's dessins d'enfants|
In 1978 Belyĭ first announced the following theorem: every algebraic curve defined over a fixed algebraic closure of the rationals can be represented as a covering of the projective line ramified over at most three points. Deligne then studied the fundamental group of the projective line minus three points. Grothendieck introduced combinatorial structures called dessins d'enfants. One consequence of Belyĭ's theorem is that the Galois action on the fundamental group of the projective line minus three points is faithful.
|Title: Néron models of elliptic curves|
We will attempt to do algebraic geometry over the integers. Doing this for elliptic curves will lead us to the idea of Néron models, and we will see how to get our hands on these guys explicitly.
|Title: Modular forms over arbitrary rings|
One way to understand moduli spaces of elliptic curves and related moduli problems better, is to study modular forms. With this in mind, I will define modular forms over arbitrary rings in the sense of Katz. I will also talk about p-adic modular forms in the sense of Serre and hopefully motivate the modern treatment of such modular forms.
|Title: The Theory of Heights and the Height of Theories|
I will talk about some of the different notions of height in arithmetic geometry, starting from the most naive, ending at some betterer ones.