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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
| DATE |
SPEAKER |
TOPIC |
| Feb 4 |
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| Feb 11 |
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| Feb 18 |
I. Sigal
University of Toronto, Canada/IAS |
Mathematical Questions Arising from Bose-Einstein Condensation
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| Feb 25 |
V. Jones
University of California, Berkeley |
Large N limit of random matrices, free probability and the graded
algebra of a planar algebra
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| 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.
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| 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 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.
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| Mar 18 |
No colloquium
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Spring break
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| Mar 25 |
C. Villani
IAS and ENS Lyon |
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| April 1 |
O. Savin
Columbia |
On a conjecture of De Giorgi
Abstract
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| April 8 |
I. Rodnianski,
Princeton |
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| 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.
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| April 22 |
J.M. Bismut
Université Paris-Sud |
The hypoelliptic Dirac operator
Abstract
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| April 28 |
Y. Minsky
Yale University |
Mapping class groups, relative hyperbolicity and rigidity
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