2008/09 Clay Lectures on Mathematics

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Lectures will be hosted by:
Research Institute for Mathematical Sciences (RIMS)
Kyoto University
Japan

March 2 - 5, 2009

Public Lectures

From Positive Characteristic to Stringy Moduli Spaces via Representation Theory

Roman Bezrukavnikov, MIT

Monday, March 2, 14:00-15:00

Many known applications of algebraic geometry to representation theory involve geometric objects of a quantum nature, such as the ring of differential operators which is a quantization of functions on the cotangent bundle. Such quantum objects and their relation to representation theory exist also in the modular setting, where one works modulo a fixed prime number p. It turns out that in this characteristic p world quantum objects are much closer to classical ones. For that reason versions of the traditional link between geometry and representations yield new geometric structures with unexpected connections to some notions of modern algebraic geometry motivated by quantum physics.

Introduction to Geometric Langlands Correspondence

Dennis Gaitsgory, Harvard University

Monday, March 2, 15:30-16:30

Let X be a Riemann surface, and let Jac(X) be its Jacobian. As is well-known, the fundamental group of Jac(X) equals the abelianization of the fundamental group of X. This means that there is a bijection between characters π₁ (X) → C^∗ and π₁ (Jac(X)) → C^∗. Let us regard the group C^∗ as GL₁(C), and let us ask: what would correspond to homomorphisms π₁ (X) → GL_n (X) for n > 1? The answer is that we have to replace Jac(X), which classifies line bundles on X, by the moduli space Bun_n(X) that classifies vector bundles of rank n on X. However, we will no longer deal with representations of the fundamental group of Bun_n(X), but with more intricate objects-automorphic sheaves. These ideas, and their number-theoretic counterpart, will be explained in the lecture.

Numbers, Patterns, and Primes

James Carlson, CLay Mathematics Institute

Monday, March 2, 17:00-18:00

What is mathematics? It begins with numbers: their first recorded use is by the Babylonians, who used them for accounting records, just as we do today. But much of research mathematics is the search for patterns --- in numbers, in shapes, in equations. We will talk about some special number patterns related to the prime numbers: 2, 3, 5, 7, 11, ... . The Greeks already knew that there is an infinite supply of prime numbers. Today we use prime numbers to safely get money from a cash machine instead of from a person at the bank. But there are still many patterns that we do not understand. One is the Riemann hypothesis, which is one of the $1 million Clay Millennium Prize problems.

Some New Ties between Algebraic Geometry and Representations via Derived Categories

Roman Bezrukavnikov, MIT

A version of Beilinson-Bernstein-Brylinsky-Kashiwara Localization Theorem holds in positive characteristic on the level of derived categories. Replacing differential operators on the flag variety by a quantization of other symplectic algebraic varieties we get localization results for other algebras of interest in representation theory (e.g. symplectic reflection algebras or finite W-algebras). The localization results translate standard questions about representations to questions about certain t-structures on the derived categories of coherent sheaves. In the case of the cotangent bundle, these same t-structures appear also in local geometric Langlands duality. Some recent results suggest that their numerics is controlled by quantum cohomology. The first fact allows to apply standard techniques of Kazhdan-Lusztig theory in new settings. We hope that the second one will yield new techniques applicable to the same circle of questions.

Derived Localization Theorems in Positive Characteristic and Noncommutative Resolutions of Singularities
Tuesday, March 3, 10:50 - 11:50 am

Tame Local Geometric Langlands Duality and Noncommutative Resolutions
Wednesday, March 4, 10:50 - 11:50 am

Noncommutative Resolutions, Bridgeland Stabilities and Quantum Cohomology
Thursday, March 5, 10:50 - 11:50 am

Quiver Varieties and Geometric Langlands Correspondence for Affine Lie Algebras

Hiraku Nakajima, Kyoto University

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group G_aff. They propose that the affine Grassmanian variety for the usual geometric Satake correspondence for the finite dimensional case is replaced by the Uhlenbeck compactification of the framed moduli space of G_cpt-instantons on R⁴/Z_l. Here the additional parameter l (a positive integer) corresponds to the level of representations of the Langlands dual group Ǧ_aff. When G = SL (r), the Uhlenbeck compactification is the quiver variety of type sl(l)_aff, and their conjecture follows from the theory of quiver varieties and I.Frenkel's level-rank duality.

In my talks, I will explain the Braverman-Fineklberg conjecture, the relationof quiver varieties to the representation theory of affine Lie algebras, and then to that of quantum affine algebras.

The Geometric Satake correspondence for an affine Lie algebra
Tuesday, March 3, 14:30 - 15:30

Quiver varieties and the affine Lie algebra
Wednesday, March 4, 14:30 - 15:30

Quiver varieties and the quantum toroidal algebra
Thursday, March 5, 14:30 - 15:30

Classical and Quantum Geometric Langlands Correspondence

Dennis Gaitsgory, Harvard University

The usual (function-theoretic) Langlands program aims to establish a correspondence (ideally, but not really, a bijection) between representations of the Galois group of a global field and irreducible automorphic representations.

A geometrization suggested by Drinfeld replaces the space of automorphic functions by the category of automorphic sheaves. In these lectures, by an automorphic sheaf we will understand a D-module on the module space Bun_G(X) of principal G-bundles on a given algebraic curve X.

This paves a way to formulating the ambitious "Classical Geometric Langlands Conjecture" that there is an equivalences between the derived categories of D-mod(Bun_G(X)) and QCoh(LocSyst_Ǧ(X)), where Ǧ is the langlands dual group, and LocSyst_Ǧ(X) is the moduli space of Ǧ-local systems on X.

The Quantum Geometric Langlands Conjecture is a 1-parameter deformation of the classical one. A remarkable feature is that away from the zero value of the parameter, the statement of the conjecture becomes symmetric in G and Ǧ: both D-mod(Bun_G(X)) and QCoh(LocSyst_Ǧ(X)) become replaced by appropriately defined categories of twisted D-modules.

Geometric Satake Equivalence and its Quantum Deformation
Tuesday, March 3, 16:00 - 17:00

Global Quantum Geometric Langlands Correspondence
Wednesday, March 4, 16:00 - 17:00

Chiral Categories and Local Quantum Geometric Langlands Correspondence
Thursday, March 5, 16:00 - 17:00