SCIENTIFIC PROGRAMS AND ACTIVITIES

November 21, 2014

The Inaugural
FIELDS MEDAL SYMPOSIUM
 

OCTOBER 15-18, 2012
AT THE FIELDS INSTITUTE
222 College St. Toronto (map)

FUNDAMENTALS OF THE LANGLANDS PROGRAM
In honour of Ngô Bào Châu
PROGRAM

Monday, October 15
Endoscopy

Afternoon Program

1.30-2.30

Jean-Loup 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' L-packets. 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.

2.45-3.45

Pierre-Henri 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 Arthur-Selberg 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 Langlands-Shelstad fundamental lemma.

 

Coffee Break

4.15-5.15

Diana Shelstad, Rutgers-Newark
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 Arthur-Selberg 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 web-cast)

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

9.30-10.30

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 Shimura-Taniyama 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

11.00-12.00

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
1.30-2.30

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.

2.45-3.45

Mark Kisin, Harvard University
Mod p points on Shimura varieties of abelian type

The Langlands-Rapoport 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 L-functions. We will report on the proof of the conjecture for Shimura varieties of abelian type.

 

Coffee Break

4.15-5.15

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

9.30-10.30

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 Frenkel-Langlands-Ngo, of Altug, and of Sakellaridis.

 

Coffee break

11.00-12.00

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 L-functions 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

9.30-10.30

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 S-duality of four-dimensional supersymmetric quantum gauge theories. I will give an overview of these links and give some examples.

 

Coffee Break

11.00-12.00

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 so-called Hitchin equations. This will be part historical, from their origins in dimensional reductions of the self-dual Yang-Mills equations, and will discuss their properties from various points of view: differential geometry, algebraic geometry and symplectic geometry.

Afternoon Program

1.30-2.30

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 three-dimensional superconformal field theory T(G) with OSp(4|4) symmetry. Arthur's SL_2 is a subgroup of this OSp(4|4) and mirror symmetry acts via an outer automorphism of OSp(4|4). 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 three-dimensional superconformal field theories which are basic examples of three-dimensional mirror symmetry -- or symplectic duality -- and also play a role in geometric Langlands.

2.45-3.45

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. Ben-Zvi (Texas) and inspired by structures arising in the Trace Formula and Ngo's proof of the Fundamental Lemma.

 

Coffee Break

4.15-5.15

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 S-duality 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


*The Fields Medal Symposium is endorsed by the International Mathematics Union. IMU President Ingrid Daubechies (Duke University) will participate in the public events and also give a Colloquium in the University of Toronto Department of Mathematics Wednesday afternoon.


Gold Level Sponsor Silver Level Sponsor    Bronze Level Sponsors  

James Stewart, Prof. Emeritus, McMaster University,text book author, donor of the Fields Institute Library

Edward Bierstone, Fields Institute and the University of Toronto
George Elliott, Fields Institute and the University of Toronto
John R. Gardner
Philip Siller, BroadRiver Asset Management, L.P.

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