I will give an introduction to the problem of parallel
repetition of two-prover games and its applications and related
results in theoretical computer science (the PCP theorem, hardness
of approximation), mathematics (the geometry of foams,
tiling...
Given a Hamiltonian on $T^n\times R^n$, we shall explain how the
sequence of suitably rescaled (i.e. homogenized) Hamiltonians,
converges, for a suitably defined symplectic metric. We shall then
explain some applications, in particular to symplectic...
I will introduce l-adic representations and what it means for
them to be automorphic, talk about potential automorphy as an
alternative to automorphy, explain what can currently be proved
(but not how) and discuss what seem to me the important open...
The systolic inequality says that if we take any metric on an
n-dimensional torus with volume 1, then we can find a
non-contractible curve in the torus with length at most C(n). A
remarkable feature of the inequality is how general it is: it
holds...
I will introduce Shimura varieties and discuss the role they
play in the conjectural relashionship between Galois
representations and automorphic forms. I will explain what is meant
by a geometric realization of Langlands correspondences, and...
In this expository talk, I will outline a plausible story of how
the study of congruences between modular forms of Serre and
Swinnerton-Dyer, which was inspired by Ramanujan's celebrated
congruences for his tau-function, led to the formulation of...
The "hard discs" model of matter has been studied intensely in
statistical mechanics and theoretical chemistry for decades. From
computer simulations it appears that there is a solid--liquid phase
transition once the relative area of the discs is...
The local Langlands conjecture (LLC) seeks to parametrize
irreducible smooth representations of a p-adic group G in terms of
Weil-Deligne parameters. Bernstein's theory describes the category
of smooth representations of G in terms of points on a...
This talk will be a biased survey of recent work on various
properties of elements of infinite groups, which can be shown to
hold with high probability once the elements are sampled from a
large enough subset of the group (examples of groups: linear...
This will be an introduction to special value formulas for
L-functions and especially the uses of modular forms in
establishing some of them -- beginning with the values of the
Riemann zeta function at negative integers and hopefully arriving
at...
We prove that the Cauchy problem for the Benjamin-Ono-Burgers
equation is uniformly globally well-posed in H^1 for all
"\epsilon\in [0,1]". Moreover, we show that for any T>0 the
solution converges in C([0,T]:H^1) to that of Benjamin-Ono equation
as...
I will survey the development of modern infinite cardinal
arithmetic, focusing mainly on S. Shelah's algebraic pcf theory,
which was developed in the 1990s to provide upper bounds in
infinite cardinal arithmetic and turned out to have
applications...
I will introduce two basic problems in random geometry. A
self-avoiding walk is a sequence of steps in a d-dimensional
lattice with no self-intersections. If branching is allowed, it is
called a branched polymer. Using supersymmetry, one can map...
Topological spaces given by either (1) complements of coordinate
planes in Euclidean space or (2) spaces of non-overlapping
hard-disks in a fixed disk have several features in common. The
main results, in joint work with many people, give...
In this talk we will discuss recent progresses meant as a
contribution to the GLS-project, the second generation proof of the
Classification of Finite Simple Groups (jointly with R. Lyons, R.
Solomon, Ch. Parker).
I will explain the main notions of the microlocal theory of
sheaves: the microsupport and its behaviour with respect to the
operations, with emphasis on the Morse lemma for sheaves. Then,
inspired by the recent work of Tamarkin but with really...
The basic ingredients of Darwinian evolution, selection and
mutation, are very well described by simple mathematical models. In
1973, John Maynard Smith linked game theory with evolutionary
processes through the concept of evolutionarily stable...
I will start with a review the basic notions of
Hamiltonian/symplectic vector field and of Hamiltonian/symplectic
group action, and the classical structure theorems of Kostant,
Atiyah, Guillemin-Sternberg and Delzant on Hamiltonian torus
actions...
