**Geometry of moduli spaces and representation theory**

The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, these students will have completed their first year, and in some cases, may already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

The main activity of the Graduate Summer School will be a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

Throughout its history representation theory thrived on connections to geometry, physics and number theory. Topology of singular algebraic varieties as a key tool in the subject goes back at least to 1980 when Kazhdan and Lusztig stated their famous conjecture. This idea is further developed in Geometric Langlands duality program, an area with roots in number theory and connections to quantum physics, where topology of certain moduli spaces plays a central role. More recent discoveries connect representation theory to constructions of algebraic geometry inspired by physics, such as quantum cohomology, which are also based on topology of moduli spaces. The goal of the program is to present these ideas in perspective, covering the necessary background and leading up to the recent progress and open problems.

Student preparation: We seek motivated students interested in the current techniques of geometric representation theory or questions in the neighboring fields motivated by it. Though familiarity with some of the topics listed above would be helpful, the formal prerequisites are limited to the content of standard introductory courses in algebraic geometry, topology and algebraic groups or Lie groups.

The 24th Annual PCMI Summer Session will be held **June 28 – July 18, 2015.**

**Click HERE** to apply to the Graduate Summer School program.

**2015 Organizers**

**Roman Bezrukavnikov, Massachusetts Institute of Technology; Alexander Braverman, Brown University; and Zhiwei Yun, Stanford University**

**2015 Graduate Summer School Lecturers**

**Alexander Braverman, Brown University**

*Geometric representation theory and quasi-maps into flag varieties.*

We are going to discuss various applications of the space of quasi-maps from a projective curve to (finite and affie) flag varieties. The specific plan is to discuss the following subjects:

1) Definition of quasi-maps; quasi-maps vs. semi-infinite flag varieties

2) Derived Satake equivalence and sheaves on quasi-maps' spaces

3) Geometric constructions of representations via quasi-maps' spaces; proof of Kazhdan-Lusztig and Jantzen conjectures

4) Uhlenbeck spaces and generalization of 1-3 to the affine case.

**Ngô Bảo Châu, Unviersity of Chicago**

*Perverse sheaves and fundamental lemmas*

Fundamental lemmas are certain identities in harmonic analysis which are very difficult to prove by purely analytic means. Geometry of certain moduli spaces and perverse sheaves offer a powerful method to prove them. Even though the general method is now well understood, the implementation in each case can be different, and some case difficult. We will explain the general method, and the implementation in different cases, including the fundamental lemma of Jacquet-Ye, of Jacquet-Rallis (proved by Yun), and of Langlands-Shelstad.

**Mark de Cataldo, Stony Brook University**

*Perverse sheaves and the topology of algebraic varieties*

These five lectures are an introduction to the use of perverse sheaves in geometry and in representation theory.

This will be done by analogy with other hopefully more familiar objects (such as locally constant sheaves), by means of many examples and also a bit more formally. There will be a discussion of perverse sheaves, their basic structure theorems (Jordan-Holder, intersection complexes and a bit of intersection cohomology), the way in which they appear in the topology of algebraic varieties and maps and, eventually, of some of the remarkable applications of these objects to geometry, representation theory and combinatorics.

A partial reference is the following survey: http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf

**Victor Ginzburg, University of Chicago**

*"Symplectic algebraic geometry and quiver varieties"*

**Hiraku Nakajima, Kyoto University**

*Perverse sheaves on instanton moduli spaces*

I will explain my joint paper `Instantons moduli spaces and W-algebras' with A.Braverman, M.Finkelberg. I will concentrate on the geometric part, that is a study of perverse sheaves on instanton moduli spaces. I place a particular emphasize on the hyperbolic restriction functor and stable envelop, which are our key tools, and appear also in other situations in geometric representation theory.

**Davesh Maulik, Columbia University**

*Quantum cohomology and symplectic resolutions*

The goal of these lectures will be to give a gentle introduction to equivariant quantum cohomology and related objects (quantum differential equation, etc.) We focus mainly on the case of equivariant symplectic resolutions, specifically quiver varieties, and the proven and conjectural special phenomena that occur here.

**Andrei Okounkov, Columbia University**

*"Computations in quantum K-theory"*

Abstract: The goals of these lectures will be two-fold. First, I would like to explain basic K-theoretic computations in enumerative geometry, including an introduction to K-theoretic Donaldson-Thomas invariants and a discussion of monodromy of quantum differential/difference equations.

Second, I will try to explain a larger picture into which such computations conjecturally (and, occasionally, provably) fit.

**Zhiwei Yun, Stanford University**

*Introduction to affine Springer fibers and Hitchin fibration*

Abstract: The geometry of affine Springer fibers and Hitchin fibration is crucial in Ngo's proof of the fundamental lemma. We will introduce these geometric objects and study their relationship. We will also study their cohomology groups and the symmetry they carry.

**Xinwen Zhu, California Institute of Technology**

*An introduction to affine Grassmannians*

Affine Grassmannians are fundamental objects in geometric representation theory and in the study of moduli spaces of G-bundles on algebraic curves, and become increasingly important in number theory. In this course, we will study their basic geometric properties and some applications.