Overview
The Clay Mathematics Institute is organizing a summer school in automorphic
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 (Toronto), 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
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 as we
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 lectures of 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
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.
Slides: Introduction to Shimura Varieties (.pdf)
(.ps)
- June 2-6
Background from Algebraic Groups
Instructor: Fiona Murnaghan
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
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
Stephen DeBacker
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
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
We will survey recent work on the bad reduction of Shimura
varieties. In particular, we will focus on the computation of the
local Hasse-Weil zeta functions at bad primes, and connections with
the Langlands correspondence.
An Introduction to the Fundamental Lemma
Tom Hales
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.
Analytic Aspects of Automorphic Forms
Peter Sarnak
The analytic theory of L-functions has many applications
to number theory and automorphic forms (and visa versa).We describe
some of this theory and its applications. Specifically to the Ramanujan
Conjectures and other spectral problems associated with quotients
of homogeneous spaces and to arithmetical problems such as quadratic
forms.
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 mathematical advisor or a senior mathematician.
Standard support amounts will include funds for local expenses and accommodation
plus economy travel.
Deadline for applications was February 15, 2003
Mathematical Preparation
There will be no formal prerequisites for the summer school. However,
the three longer courses will all be presented from general standpoint
of reductive algebraic groups. This is the reason for the initial one
week course, devoted to the statement and description of some of the
basic properties of algebraic groups. Since this course will not include
proofs, participants are encouraged to come with some prior understanding
of the subject. A familiarity with any of the topics in the following
references would be a definite asset to bring to the summer school.
A. Borel, Linear Algebraic Groups, Benjamin, 1969.
J. Humphreys, Introduction to Lie Algebras and Representation Theory,
Springer-Verlag, 1972.
T. Springer, Linear Algebraic Groups, Birkhäuser, 1981.
J.-P. Serre, Complex Semisimple Lie Algebras, Springer-Verlag,
1987.
Participants will be assumed to have some knowledge of number theory.
The main theorems
of class field theory will be reviewed in the course on Shimura varieties,
but again without proof. A good general reference is,
J. Cassels, and A. Fröhlich, Algebraic Number Theory, Thompson,
1967.
The thesis of Tate, reprinted in this volume, is especially recommended,
for its introduction to adèles, and its construction of abelian
L-functions.
A good introductory reference to the general theory of automorphic forms
is the proceedings of the Edinburgh instructional conference:
T.N. Bailey and A.W. Knapp, Representation Theory and Automorphic
Forms, Proc. Sympos. Pure Math., 61, Amer. Math. Soc., 1996.
The Clay Mathematics Institute (CMI) is a private, non-profit foundation,
dedicated to increase and to disseminate mathematical knowledge.
The primary objectives and purposes of The Clay Mathematics Institute
(CMI) are, to increase and disseminate mathematical knowledge, to educate
mathematicians and other scientists about new discoveries in the field
of mathematics,to encourage gifted students to pursue mathematical careers,
and to recognize extraordinary achievements and advances in mathematical
research.
Applications to Attend
The final date to register was February 15, 2003, therefore we are
no longer accepting applications.
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Thematic
Program on Automorphic Forms