Graduate Algebraic Geometry Seminar Fall 2011

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Fall 2011

Wednesdays 4:30pm-5:30pm, B309 Van Vleck

The purpose of this seminar is to have a talk on each Wednesday by a graduate student to help orient ourselves for the Algebraic Geometry Seminar talk on the following Friday. These talks should be aimed at beginning graduate students, and should try to explain some of the background, terminology, and ideas for the Friday talk.

Give a talk!

We need volunteers to give talks this semester. If you're interested contact David. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.

Fall 2011 Semester

Date	Speaker	Title (click to see abstract)
September 14 (Wed.)	Lalit Jain	Introduction
September 21 (Wed.)	David Dynerman	Artin Stacks
September 28 (Wed.)	Nathan Clement	Hodge Theory and the Frobenius Endomorphism
October 5 (Wed.)	Jeff Poskin	Fine Moduli Spaces and Hilbert Schemes
October 12 (Wed.)	Ahmet Kabakulak	Resolution of Singularities and Nash Mappings
October 19 5:15pm (Wed.)	Nathan Clement	Rise over Run, a Gentle Introduction to Derived Categories
October 26 4:45pm (Wed.)	Daniel Ross	Introduction to the Riemann-Hilbert correspondence

September 14

Lalit Jain

Title: Introduction

Abstract: In this talk I'll present a few of the basic concepts of scheme theory and provide several motivating examples.Schemes are a fundamental object in modern algebraic geometry that greatly generalize varieties. The target audience is new graduate students who have had no (or perhaps only a classical) introduction to algebraic geometry.

September 21

David Dynerman

Title: Artin Stacks

Abstract: This is a preparatory talk for Yifeng Lui's seminar talk. Yifeng will be talking about recent developments in defining sheaves on Artin stacks, so I will attempt to define an Artin stack and hopefully work out an example or two.

September 28

Nathan Clement

Hodge Theory and the Frobenius Endomorphism: The curious tale of calculus in characteristic p>0.

Abstract: I will present the basic ideas neccesary to understand (1) the original statement of Hodge decomposition and the idea of the proof and (2) the proof given by Pierre Deligne and Luc Illusie of the analagous statement on schemes of characteristic p>0 given some special lifting condition. A brief argument extends the result to schemes of characteristic 0. If time permits, I will give an idea of what Professor Caldararu's more geometric take on the situation might be.

October 5

Jeff Poskin

Fine Moduli Spaces and Hilbert Schemes

Abstract:

October 12

Ahmet Kabakulak

Resolution of Singularities and Nash Mappings

Abstract: This is a preparatory talk for Javier Fernandez de Bobadilla's talk on Nash problem for surfaces. Nash Problem is about surjectivity of the `Nash Mapping'. Injectivity is showed by J.F. Nash. Surjectivity for surfaces (over an algebraically closed field of char 0) is showed by the presenter. I will try to explain `resolution of singularities' over an algebraic variety X, the `space of arcs' in the variety and the mapping (which is called the Nash mapping) between these two objects.

October 19

Nathan Clement

Special time: 5:15pm

Rise over Run, a Gentle Introduction to Derived Categories

Abstract: In this talk I will start by defining the derived category of an abelian category. I will give some examples of constructions in the derived category and I will explain the idea of derived self-intersection.

October 26

Daniel Ross

Special time: 4:45pm

Introduction to the Riemann-Hilbert correspondence

Abstract: Let X = C \ S be C without finitely many points. For a system D of n differential equations on X, written as F' = A(x)F with F an n x n matrix (which we additionally demand to be invertible) and with A singular (but not too badly so) on S, the solutions of D are naturally viewed as holomorphic functions on the universal cover Y of X. Then the deck transformations of Y -> X clearly act on the space of solutions, and for F_0 a fixed solution, any other solution is given as s(F_0), (which we can write as rho(s)*F_0) for some s in the deck group. This defines the monodromy representation rho_D : pi_1(X) -> End(C^n) corresponding to D. Hilbert's 21st problem asks whether we can reverse this to get an honest correspondence between the topological information of representations of pi_1(X) and the analytic information of systems of differential equations. In our talk we will introduce this problem in more detail and then discuss the generalisation of this problem to a singular variety X using holonomic D-modules with the goal of being able to gesture towards the modern formulation of the above `Riemann-Hilbert correspondence' as an equivalence between appropriate topological and analytic categories associated to X.