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Conférenciers pléniers
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- ROBERT GURALNICK, University of Southern California
Probabilistic Methods in Finite Group Theory
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We shall discuss the use of probabilistic methods in generation of
finite groups. We shall give a survey of some of the main results in
the area and also discuss some more recent results. We shall also
discuss some related results on derangements in finite primitive
permutation groups.
- UFFE HAAGERUP, South Denmark University
Free probability and the invariant subspace problem for von
Neumann algebras
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Does every operator T on a Hilbert space H have a non-trivial
closed invariant subspace? This is the famous and still open
"invariant subspace problem" for operators on a Hilbert space. A
natural generalization of the problem is: Let M be a von Neumann
algebra on a Hilbert space H. Does every operator T in M have a
non-trivial closed invariant subspace K affiliated with M? (K
is affiliated with M, iff the orthogonal projection on K belongs
to M.) In the special case, when M is a II1-factor
(i.e., a infinite dimensional von Neumann factor with a bounded
trace), it turns out, that "almost all" operators in M have
non-trivial closed invariant subspaces affiliated with M. More
precisely, it holds for all operators in M for which L. G. Brown's
spectral distribution measure for T is not concentated in a single
point of the complex plane. The result is obtained in collaboration
with Hanne Schultz, and it relies in a crucial way on Voiculescu's
free probability theory.
- BRYNA KRA, Northwestern University
Additive Combinatorics and Ergodic Theory
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A beautiful result in combinatorial number theory is Szemeredi's
Theorem: a set of integers with positive upper density contains
arbitrarily long arithmetic progressions. In the 1970s, Furstenberg
established the deep connections between combinatorics and ergodic
theory, using ergodic theory to prove Szemeredi's Theorem. This
development lead to the field of Ergodic Ramsey Theory and many new
combinatorial and number theoretic statements were proven using
ergodic theory. In the last year, this interaction took a new twist,
with ergodic methods playing an important role in Green and Tao's
proof that the prime numbers contain arbitrarily long arithmetic
progressions. I will give an overview of this interplay, with a focus
on recent developments in ergodic theory.
- ANDREW MAJDA, New York University, Courant Institute, 251 Mercer St., NY,
NY 10012
New waves, PDEs, and coarse-grained stochastic lattice
models for the tropics
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One of the most important and prototype multi-scale systems involves a
comprehensive model for the coupled atmosphere ocean system for both
climate change and longer term weather prediction. One of the
striking recent observational discoveries is the profound impact of
variations in the tropics on all of these issues. The talk has four
parts:
(1) an introduction to these issues;
(2) novel behavior of waves in the simplest tropical climate
models and their mathematical analysis compared/contrasted with
relaxation limits and combustion waves;
(3) systematic mathematical strategies for coarse graining
stochastic lattice models for both material science and climate
physics;
(4) application of (3) to show the dramatic effect in simple
tropical climate models of stochastic effects on both the
climatology and fluctuations.
- ODED SCHRAMM, Microsoft
Understanding 2D critical percolation: from Harris to Smirnov
and beyond
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There are numerous predictions from statistical physics regarding
random systems in the plane which were until recently beyond the reach
of mathematical understanding. Some of the better-known examples
include percolation and the Ising model. We will focus on percolation
and describe our growing understanding of it through a sequence of
insights (which are simple in hindsight) from 1960 through today.
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