2003 Summer School

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Overview

The Clay Math Institute is sponsoring a summer school in automophic forms in June, 2003. The school will be held at the Fields Institute in Toronto and will be aimed at graduate students and mathematicians within five years of their Ph.D.
The school will begin with three weeks of foundational courses centered around the trace formula: one course on the statement and proof of the trace formula, two courses providing background material on reductive groups and harmonic analysis on those groups, and a fourth course on Shimura varieties, which provide an illuminating application of the trace formula. The fourth week will consist of five short courses on more specialized topics related to the main themes of the school. While there are no formal prerequisites, preference will be given to applicants with some prior knowledge of algebraic groups or number theory.

Organizing Committee

James Arthur (Princeton), David Ellwood (Boston & CMI), Robert Kottwitz (Chicago)

Summer School Lecture Courses: June 2 - June 20

• June 2 - 20
  Introduction to the Trace Formula
  Instructor: James Arthur (Princeton)

The topic of this course will be the global trace formula for a reductive group over a number field. We shall begin with a brief overview of the subject, taking motivation from the case of compact quotient. We shall then prove as much of the general formula aswe can. In the process, we shall introduce the orbital integrals and characters, and their weighted variants, that are the main terms in the trace formula. The deeper study of these local objects will be the subject of the course of Kottwitz, and the lecturesof DeBacker and Hales in the final week. General applications of the trace formula will actually require two successive refinements, the invariant trace formula and the stable trace formula. If time permits, we shall discuss these refinements, and the local problems of comparison whose solutions are required for applications.

• June 2- 20
  Introduction to Shimura Varieties
  Instructor: James Milne (Michigan)
  
  Shimura varieties are the natural generalization of elliptic modular curves. Examples include the Hilbert modular varieties and the Siegel modular varieties. The fundamental theorem in the theory of Shimura varieties is the existence and uniqueness of canonical models over number fields. A primary goal of the course will be to obtain a good understanding of this theorem. In particular, we shall discuss the theorem of Shimura and Taniyama on complex multiplication, and the various ways of realizing Shimura varieties as moduli varieties. We expect also to include the following two topics: the structure of Shimura varieties modulo p, especially in the PEL case; boundaries of Shimura varieties and their various compactifications.
  
• June 2-6
  Background from Algebraic Groups
  Instructor: Fiona Murnaghan (Toronto)
  
  We will describe, without giving proofs, some of the main results in the theory of algebraic groups. The main emphasis will be on the classification and stucture of reductive algebraic groups.


• June 9-20
  Harmonic Analysis on Reductive Groups and Lie Algebras
  Instructor: Robert Kottwitz (Chicago)
  
  This course will introduce the basic objects of study in harmonic analysis on reductive groups and Lie algebras over local fields: orbital integrals, their Fourier transforms (in the Lie algebra case) and characters of irreducible representations (in the group case). The emphasis will be on p-adic fields and the Lie algebra case (to which the group case can often be reduced using the exponential map). Some of the main theorems involving these objects will be discussed: Howe's finiteness theorem; Shalika germs, the local character expansion and its Lie algebra analog; local integrability of Fourier transforms of orbital integrals; and the Lie algebra analog of the local trace formula.
  

Advanced Short Courses: June 23- June 27

List of Lecturers:

An Introduction to Homogeneity with applications)
Steven DeBaker (Harvard)

In the early 1990s, J.-L. Waldspurger established a very precise version of Howe's finiteness conjecture (for the Lie algebra). We shall discuss this result and some of its applications to harmonic analysis on reductive p-adic groups.

Geometry and topology of compactifications of modular varieties
Mark Goresky (Princeton),

We will describe the construction, basic properties and applications of the Baily-Borel (Satake) compactification, the Borel-Serre and reductive Borel-Serre compactifications, and the toroidal compactifications.

Bad reduction of Shimura varieties
Thomas Haines (Maryland),

A collection of conjectural identities between integrals on reductive groups has become known as the "Fundamental Lemma." These lectures will describe these conjectural identities, and will discuss the progress that has been made toward their proof.

An introduction to the fundamental lemma
Tom Hales (Pittsburg),

TBA
Peter Sarnak (Princeton).

Applying to attend

Interested participants should apply to attend the summer school by sending in the application form available here.

Graduate and Postdoctoral Funding

Funding is available to graduate students and postdoctoral fellows (within 5 years of their PHD) to attend the summer school. We anticipate that funding will be available for 90 graduate students and young mathematicians. Interested candidates must forward with their application, a letter of recommendation from their mathematicic advisor or a senior mathematician.
Standard support amounts will include funds for local expenses and accommodation plus economy travel.
Deadline for applications isFebruary 15, 2003

Mail list

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For more information please contact automorphic@fields.utoronto.ca