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Spring 2009

WEDNESDAY 4:30-5:30 pm
Tea served at 4:00 p.m.
Fine Hall 314

Information for the speakers

Feb 4

Feb 11

Feb 18 I. Sigal
University of Toronto, Canada/IAS
Mathematical Questions Arising from Bose-Einstein 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 Diaconis-Fulton addition to a free boundary problem
Start with n particles at each of k points in the d-dimensional 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 free-boundary 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
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 half-plane 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 equi-distributed. 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
On a conjecture of De Giorgi
April 8 I. Rodnianski,

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 power-like (polynomial) speed. There is no general theory of parabolic dynamics and a few classes of examples are relatively well-understood: area-preserving 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 power-like (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 non-renormalizable 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 Paris-Sud
The hypoelliptic Dirac operator
April 28 Y. Minsky
Yale University
Mapping class groups, relative hyperbolicity and rigidity

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