The local type of difference equations
Algebra and Algebraic Geometry seminar, 03/02/2018
Abstract: The theory of algebraic differential equations on the affine line is very well-understood. In particular, there is a well-defined notion of restricting a D-module to a formal neighborhood of a point, and these restrictions are completely described by two vector spaces, called vanishing cycles and nearby cycles, and some maps between them. We give an analogous notion of "restriction to a formal disk" for difference equations that satisfies several desirable properties: first of all, a difference module can be recovered uniquely from its restriction to the complement of a point and its restriction to a formal disk around this point. Secondly, it gives rise to a local Mellin transform, which relates vanishing cycles of a difference module to nearby cycles of its Mellin transform. Since the Mellin transform of a difference module is a D-module, the Mellin transform brings us back to the familiar world of D-modules.
Hilbert's twenty-first problem and the Riemann-Hilbert Correspondence
Algebra and Combinatorics seminar at ICMAT, 06/16/2016
Abstract: Given a differential equation in the complex numbers with regular singular points, we can ask how their solutions change when they are extended analitically around the singular points (for example, the solution to the equation y'=1/x, the logarithm, changes by 2πi when we go around the origin). Hilbert's twenty-first problem asks about the inverse construction: given the information about the monodromy, that is, about the change around singular points, is there a linear differential equation with this monodromy? The answer (given by Plemelj, Ilyashenko, Bolibrukh, Kostov and others) is "almost always yes, but not always". I won't explain every result, but the construction serves as an excuse for motivating several objects in algebraic geometry: sheaves, vector bundles, connections, modifications of bundles and the Riemann-Hilbert correspondence.
Computing the cohomology of line bundles on abelian varieties
Abelian varieties reading group, 03/14/2016
Abstract: Kempf's "Complex Abelian Varieties and Theta Functions", sections 3.4 to 3.6. Here's my notes.
Fun with commutative groups
Graduate Algebraic Geometry Seminar, 02/14/2018 and 02/21/2018
Week 1 abstract: 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.
Week 2 abstract: 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.
Vector bundles on the projective line and on elliptic curves
Graduate Algebraic Geometry Seminar, 09/13/2017 and 09/27/2017
Week 1 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.
Week 2 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.
Symplectic geometry
Graduate Algebraic Geometry Seminar, 10/28/2015 and 11/04/2015
Abstract:
When dynamics get hectic
For reasons symplectic
Don't sit and brute-force them all day.
Find nice functions and list them -
Integrable system!
Let symmetries show you the way.
(We owe this poem to Ed Dewey)
The natural numbers form a field
AMS Student Chapter Seminar, 10/14/2015
Abstract: But of course, you already knew that they form a field: you just have to biject them into Q and then use the sum and product from the rational numbers. However, out of the many field structures the natural numbers can have, the one I'll talk about is for sure the cutest. I will discuss how this field shows up in "nature" (i.e. in the games of some fellows of infinite jest) and what cute properties it has.
Winning games and taking names
AMS Student Chapter Seminar, 01/28/2015
Abstract: So let's say we're already amazing at playing one game (any game!) at a time and we now we need to play several games at once, to keep it challenging. We will see that doing this results in us being able to define an addition on the collection of all games, and that it actually turns this collection into a Group. I will talk about some of the wonders that lie within the group. Maybe lions? Maybe a field containing both the real numbers and the ordinals? For sure it has to be one of these two!
Dessins d'enfants
Graduate Algebraic Geometry Seminar, 03/11/2015
Abstract: I will talk about the definition of dessins d'enfants, both as combinatorial objects and as algebraic curves with a Belyi function, on which the Galois group of the rational numbers acts on a pretty (yet mysterious) way. I will describe this action, and how we get an embedding of the Galois group into the Grothendieck-Teichmüller group. If time permits, I will say more things about this action. No scheme theory is required for the talk, as a very small epsilon is known by the speaker. However, some topology will be used, but I will try to state all the theorems needed.
How to make it as a Hackenbush player in the planet Zubenelgenubi 4
Madison Math Circle, 10/02/2017
Abstract: In the distant planet of Zubenelgenubi 4, we live our life without numbers. I know, how do we pass our time if we can't construct a smartphone without numbers? The answer is that we have invented an extremely violent sport about chopping down trees called Hackenbush, and playing this game is an essential social skill in Zubenelgenubi 4. I will teach you how to play the pen and paper version of Hackenbush, and hint at how learning this game leads to a kind of math that is highly illegal in 254,233 planetary systems.