Annual Report 2005 - Fields Institute - University of Toronto

Many speakers reported on recent progress made on the
Langlands Program. Several speakers emphasized the trace
formula approach, discussing topics such as stabilization
of the twisted trace formula and norm maps, explicit trace
identities, and analytic applications of trace formulas.
Robert Langlands’ talk on trace formulas and number
theory drew a large audience, including many Torontoarea
researchers and graduate students. Functoriality was
a theme of some talks, namely functoriality for unitary
groups, functoriality related to Arthur’s conjectures
describing representations of real and p-adic groups that
occur in spaces of automorphic forms, and relations
between functoriality and the Ramanujan and Selberg
conjectures. The subject of endoscopy came up in talks on
endoscopic decompositions of local L-packets associated to
certain depth zero supercuspidal representations, and also
on character identities related to twisted endoscopy.
Bao Chau Ngo and Gérard Laumon discussed their joint
work on the proof of the fundamental lemma for unitary
groups over local fields of positive characteristic. Talks on
Shimura varieties focused on recent progress on local models
of such varieties, and questions concerning Shimura
varieties and parahoric Hecke algebras.
Other topics discussed included simplification of reduction
theory and Arthur’s truncation operator using buildings;
questions of stability related to properties of characters
of depth zero supercuspidal representations of reductive
p-adic groups, and the problem of determining which
Langlands parameters correspond to unitary representations.
After the end of the first day’s talks, the conference participants
attended a reception at Hart House, sponsored by
the University of Toronto Mathematics Department. The
conference reception was held at the Faculty Club on the
evening of the third day of the conference, immediately
before the banquet, which was also held there. Everyone
at the banquet enjoyed Steve Gelbart’s display of slides of
photos featuring Jim and his wife Penny, dating back to the
1970’s. Several people took the opportunity to make some
comments, perhaps recalling the circumstances of their
first meeting with Jim, or telling a story involving Jim.
Funding was provided by the Clay Mathematics Institute,
the Fields Institute, the MSRI-network conference fund, the
U.S. National Science Foundation, and the Mathematics
Department, the Faculty of Arts and Science, and the Connaught
Committee at the University of Toronto.
G e n e r a l S c i e n t i f i c A c t i v i t i e s
Speakers:
Joseph Bernstein (Tel Aviv)
Subconvexity estimates and representation theory
William Casselman (UBC)
Truncation and buildings
Laurent Clozel (Paris South)
Arthur’s conjectures, restriction principle, and mysterious
functorialities
Stephen DeBacker (Michigan)
Murnaghan-Kirillov theory for depth zero supercuspidal
representations
Thomas Haines (Maryland)
Shimura varieties with parahoric level structure
David Kazhdan (The Hebrew University)
On endoscopic decomposition of certain weight zero representations
Henry Kim (Toronto)
Functoriality for unitary groups
Robert Kottwitz (Chicago)
Arthur’s work on the trace formula
Jean-Pierre Labesse (Paris 7)
Norm map for twisted endoscopy
Robert Langlands (IAS)
The trace formula and the theory of numbers
Erez Lapid (The Hebrew University)
Explicit trace identities
Gerard Laumon (Paris, Orsay)
The fundamental lemma for unitary groups (I)
Werner Muller (Bonn)
The trace formula and spectral theory of automorphic forms
Bao Chau Ngo (Paris 13)
The fundamental lemma for unitary groups (II)
Michael Rapoport (Bonn)
Local models of Shimura varieties
Fields Institute 2005 ANNUAL REPORT 67