

Spring 2009
WEDNESDAY 4:305:30 pm
Tea served at 4:00 p.m.
Fine Hall 314
Information for the speakers
DATE 
SPEAKER 
TOPIC 
Feb 4 


Feb 11 


Feb 18 
I. Sigal
University of Toronto, Canada/IAS 
Mathematical Questions Arising from BoseEinstein Condensation

Feb 25 
V. Jones
University of California, Berkeley 
Large N limit of random matrices, free probability and the graded
algebra of a planar algebra

Mar 4 
Y. Peres
University of California, Berkeley 
Internal aggregation Models: From DiaconisFulton addition to a free boundary problem
Start with n particles at each of k points in the ddimensional lattice, and let each particle perform simple random walk until it reaches an unoccupied site. The law of the resulting random set of occupied sites does not depend on the order in which the walks are performed, as shown by Diaconis and Fulton. We prove that if the distances between the starting points are suitably scaled, then the set of occupied sites has a deterministic scaling limit. In two dimensions, the boundary of the limiting shape is an algebraic curve of degree 2k. (For k = 1 it is a circle, as proved in 1992 by Lawler, Bramson and Griffeath.) The limiting shape can also be described in terms of a freeboundary problem for the Laplacian and quadrature identities for harmonic functions. I will describe applications to the abelian sandpile, and show simulations that suggest intriguing (yet unproved) connections with conformal mapping. Joint work with Lionel Levine.

Mar 11 
K. Soundararajan
Stanford 
Quantum Unique Ergodicity and Number Theory
A fundamental problem in the area of
quantum chaos is to understand the distribution of
high eigenvalue eigenfunctions of the Laplacian
on certain Riemannian manifolds. A particular
case which is of interest to number theorists
concerns hyperbolic manifolds arising as a
quotient of the upper halfplane by a discrete
``arithmetic" subgroup of SL_2(R) (for example,
SL_2(Z), and in this case the corresponding eigenfunctions are
called Maass cusp forms). In this case, Rudnick and Sarnak have
conjectured that the high energy eigenfunctions
become equidistributed. I will discuss
some recent progress which has led to a
resolution of this conjecture, and also on a holomorphic analog
for classical modular forms.

Mar 18 
No colloquium

Spring break

Mar 25 
C. Villani
IAS and ENS Lyon 

April 1 
O. Savin
Columbia 
On a conjecture of De Giorgi
Abstract

April 8 
I. Rodnianski,
Princeton 

April 15 
G. Forni
University of Maryland 
Invariant distributions and scaling in parabolic dynamics
A smooth dynamical system is often called parabolic if nearby orbits
diverge with powerlike (polynomial) speed. There is no general theory
of parabolic dynamics and a few classes of examples are relatively
wellunderstood: areapreserving flows with saddle singularities on
surfaces (or, equivalently, interval exchange transformations) and
to a lesser extent 'rational' polygonal billiards; SL(2,R) unipotent
subgroups (horocycle flows on surfaces of constant negative curvature)
and nilflows. In all the above cases, the typical system is uniquely
ergodic, hence ergodic averages of continuous functions converge unformly
to the mean. A fundamental question concerns the speed of this convergence
for sufficiently smooth functions. In many cases it is possible to prove
powerlike (polynomial) upper bounds. A unified approach to this problem
consists in constructing invariant distributions (in the sense of L. S.
Sobolev or L. Schwartz) usually by methods of harmonic analysis and studying
how they rescale under an appropriate 'renormalization' scheme. This
approach yields quite precise bounds for many of the above examples but
often cannot be implemented for lack of an (effective) renormalization.
In this talk, after a review of some of the main known results for
renormalizable systems, we will present a quantitative equidistribution
result for some nonrenormalizable nilflows and we will discuss some new
ideas
we have introduced (in joint work with L. Flaminio) to deal with this
problem. Bounds on Weyl sums that can be derived from our results will be
discussed.

April 22 
J.M. Bismut
Université ParisSud 
The hypoelliptic Dirac operator
Abstract

April 28 
Y. Minsky
Yale University 
Mapping class groups, relative hyperbolicity and rigidity


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