Monday,
October 15
Endoscopy 
Afternoon
Program

1.302.30

JeanLoup
Waldspurger, Institut de mathématiques de
Jussieu
Introduction to endoscopy
We
will present the basic definitions of the theory of
endoscopy. For a reductive group defined over a local
field, endoscopy is useful to describe the Langlands'
Lpackets. For a reductive group defined over a number
field, it is useful to describe the multiplicities
of automorphic representations in the discrete spectrum.
We will try to
explain these descriptions and to give some informations
about the conjectural method of proof, due to Langlands,
that is the stabilization of the trace formula.

2.453.45

PierreHenri
Chaudouard,Institut de Mathématiques de Jussieu
et Université Paris 7
Endoscopy and the geometry of the Hitchin fibration
In
this talk (based on the work of Ngô Bao Châu
and on my joint work
with Gérard Laumon) we will give an overview
of the interplay between endoscopy, the ArthurSelberg
trace formula and the geometry of the Hitchin fibration.
More precisely, we will try to explain how a deep
cohomological property of the Hitchin fibration is
the key to get various identities between orbital
integrals, including the LanglandsShelstad fundamental
lemma.


Coffee
Break

4.155.15

Diana
Shelstad, RutgersNewark
Transfer in Endoscopy (and beyond) for Real Groups
We
consider real reductive groups and describe some theorems
on endoscopic transfer in this setting. In preparation
we review the notion of stabilization first from a
more elementary perspective and then briefly from
the global perspective of the ArthurSelberg trace
formula. If time permits, we also discuss very briefly
the stable transfer for orbital integrals on real
groups envisaged by Langlands within the theme of
Beyond Endoscopy and describe the complementary nature
of the two transfers via examples we carry throughout
the talk.

Evening
Program
Public Opening at the Isabel
Bader Theatre,93 Charles St., University of Toronto
(map)
(Click here to view the
webcast)

7:00
p.m.

Welcome,
and Public Opening with:
Ingrid Daubechies, President of the International
Mathematical Union
His Excellency Le Sy Vuong Ha, Ambassador of Vietnam
to Canada
The Honourable Bob Rae, Leader of Liberal Party of Canada, Member of Parliament for Toronto Centre, former Premier of Ontario
The Honourable Glen Murray, Minister of Training,
Colleges and Universities, Government of Ontario

Public
Lecture: The Langlands Program: Number Theory, Geometry and the Fundamental Lemma
James Arthur, University of Toronto
Public
Lecture: The Fundamental Lemma
Ngô
Bào Châu, University of Chicago

Ingrid Daubechies

James Arthur

Ngo Bao Chau 
Tuesday,
October 16
Shimura Varieties and Galois Representations 
Morning
Program

9.3010.30

Richard
Taylor, Institute for Advanced Study, Princeton
Reciprocity laws (slides)
Reciprocity
laws provide a rule to count the number of solutions
to a fixed polynomial equation, or system of polynomial
equations, modulo a variable prime number. The rule
will involve very different objects: automorphic forms
and discrete subgroups of Lie groups. The prototypical
example is Gauss' law of quadratic reciprocity, which
concerns a quadratic equation in one variable. Another
celebrated example is the ShimuraTaniyama conjecture
which concerns a cubic equation in two variables.
This
will be a colloquium style talk aimed at a general
mathematical audience, and not at number theorists.
I will start with Gauss' law and work my way up to
somewhat more complicated examples. At the end of
the talk I hope to indicate the current state of our
knowledge and the role Ngo's work played in getting
there.


Coffee
Break

11.0012.00

Michael
Harris, Institut de mathématiques de Jussieu
Shimura varieties and the search for a Langlands
transform
The
Langlands reciprocity conjectures predict the existence
of a correspondence between certain classes of representations
of Galois groups of number fields and automorphic
representations. The study of the geometry of Shimura
varieties has been central in establishing these conjectures
in the cases where they are known. The talk will explain
how Shimura varieties provide a link between automorphic
forms and Galois theory, and will review some of the
most recent results. Attention will be paid to the
role of the Fundamental Lemma in constructing cases
of the Langlands correspondence.

Afternoon
Program

1.302.30

Sophie
Morel, Princeton University
The cohomology of Shimura varieties at unramified
places
In
this talk, I will explain the usual way to calculate
the cohomology of Shimura varieties at places of good
reduction.

2.453.45

Mark
Kisin, Harvard University
Mod p points on Shimura varieties of abelian type
The
LanglandsRapoport conjecture gives a description
of the mod p points of a Shimura variety with hyperspecial
level structure at p. The ultimate motivation for
the conjecture is Langlands' program to describe the
zeta function of a Shimura variety in terms of automorphic
Lfunctions. We will report on the proof of the conjecture
for Shimura varieties of abelian type.


Coffee
Break

4.155.15

Peter
Scholze, Mathematisches Institut der Universität
Bonn
The cohomology of Shimura varieties at ramified places
Starting
with ideas of Langlands, there is a conjectural endoscopic
description of the cohomology of Shimura varieties
with its Galois action. In this talk, we will explain
the results of many people that verified many cases
of this conjecture using the fundamental lemma, with
particular emphasis on the places of bad reduction.

Michael Harris

Sophie Morel

Mark Kisin

Peter Scholze

Evening
Program


Special
Program for High School and Undergraduate Students

Wednesday,
October 17
Beyond Endoscopy 
Morning
Program

9.3010.30

Ngô
Bào Châu, University of Chicago
Recent works inspired by Langlands' paper "Beyond
endoscopy"
In
the above mentioned paper, Langlands introduced a
new way to use the trace formula to extract information
about his functoriality conjecture. I will survey
on recents works directly inspired by it, including
works of FrenkelLanglandsNgo, of Altug, and of Sakellaridis.


Coffee
break

11.0012.00

Peter Sarnak, Institute for Advanced Study, Princeton
Some analytic applications of the trace formula before
and beyond endoscopy
We
describe briefly some of the ways in which the trace
formula has been used in a non comparative way. In
particular we focus on families of automorphic Lfunctions
symmetries associated with them which govern the distribution
of their zeros.We highlight some recent general results
and also the use of the function field in establishing
critical combinatorial identities in the number field.

Afternoon
Program


Mid
Symposium break

5:30

Panel
Discussion on Women in Mathematics

Evening


Banquet

Thursday,
October 18
Geometric Langlands Program and Mathematical Physics 
Morning
Program

9.3010.30

Edward Frenkel, University of California, Berkeley
An overview of the geometric Langlands Program
The
Langlands Program was launched in the late 1960s with
the goal of relating Number Theory and Harmonic Analysis.
In the last 20 years a geometric version has been
developed, in which familiar objects from these two
fields are replaced with their geometric analogues.
One is then led to a correspondence, or duality, between
these objects. The most satisfying form of this correspondence
is inherently categorical: it relates categories of
sheaves attached to a complex algebraic curve and
two Langlands dual groups. This form of the Langlands
correspondence turns out to be closely linked to the
Sduality of fourdimensional supersymmetric quantum
gauge theories. I will give an overview of these links
and give some examples.


Coffee
Break

11.0012.00

Nigel Hitchin, Oxford University
Higgs bundles, past and present
The
talk will be an overview of the moduli spaces of Higgs
bundles, or equivalently solutions to the socalled
Hitchin equations. This will be part historical, from
their origins in dimensional reductions of the selfdual
YangMills equations, and will discuss their properties
from various points of view: differential geometry,
algebraic geometry and symplectic geometry.

Afternoon
Program

1.302.30

Edward Witten, Institute for Advanced Study,
Princeton
Superconformal
Field Theory And The Universal Kernel of Geometric Langlands
The
universal kernel of geometric Langlands for a group
G can be understood as a threedimensional superconformal
field theory T(G) with OSp(44) symmetry. Arthur's
SL_2 is a subgroup of this OSp(44) and mirror symmetry
acts via an outer automorphism of OSp(44). As an
application, we will explain a quantum field theorist's
view of " geometric Eisenstein series.'' T(G)
is the prototype of a family of threedimensional
superconformal field theories which are basic examples
of threedimensional mirror symmetry  or symplectic
duality  and also play a role in geometric Langlands.

2.453.45

David Nadler, Northwestern University
Traces and loops
We
will begin with a mathematical orientation to 4D topological
field theory and specifically the Geometric Langlands
Program. Within this framework, our focus will be
on the intimate relation between traces of Hecke operators
and loop spaces. This naturally leads to enhancements
of characters to objects of categories and orbital
integrals to calculations of cohomology. Applications
include dualities for Lusztig's character sheaves
and a developing theory of character sheaves for loop
groups. Much of the talk will be based on joint work
with D. BenZvi (Texas) and inspired by structures
arising in the Trace Formula and Ngo's proof of the
Fundamental Lemma.


Coffee
Break

4.155.15

Tamás Hausel, École Polytechnique
Fédérale de Lausanne and Mathematical
Institute, Oxford
Mirror
symmetry in the character table of SL_n(F_q)
In
this talk I will start by giving an overview of how
ideas from Sduality in mathematical physics lead
to expectations in arithmetic harmonic analysis of
both the fundamental lemma, as proved by Ngo Bao Chau,
and mirror symmetry considerations for the character
table of SL_n(F_q). The latter we manage to prove
by certain twisted character formulas and careful
combinatorial study of the character table. This is
joint work with Martin Mereb and Fernando Rodriguez
Villegas.

Edward Frenkel

Nigel Hitchin

Edward Witten 
David Nadler 
Tamas Hausel
