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DEPARTMENT
COLLOQUIUM

SPRING 2007

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


Information for the speakers

DATE
SPEAKER
TOPIC
Feb 7 Scott Sheffield
NYU and IAS
Random two-dimensional geometry
The two-dimensional Gaussian free field (GFF) is a natural two-dimensional-time analog of Brownian motion. The Schramm-Loewner evolution (SLE) is a type of random planar path which has been widely studied in recent years due to its role in conformal field theory and statistical physics.
We exhibit some surprising connections between SLE and the GFF. Specifically, we use the GFF to "perturb" Euclidean geometry in order to construct an everywhere-singular connection (the "AC geometry") whose autoparallels are forms of SLE. This analysis enables us to settle some conjectures about SLE. See math.nyu.edu/faculty/sheff/spokes.html for a graphical description of what an AC geometry is.
Feb 14 Cedric Villani
Ecole Normale Superieure de Lyon

Feb 21 Sylvia Serfaty
NYU
A game interpretation of curvature flows and other nonlinear PDE's
Mean curvature flow is the motion of a curve (or hupersurface) with normal velocity equal to its mean curvature. In a joint work with Robert Kohn, we showed how an elementary two-person deterministic game converges to the viscosity solution of the mean curvature equation (in level set formulation). This gives a parallel to the optimal control interpretation of first order Hamilton-Jacobi equations. In recent progress, we are able to find analogous interpretations for all parabolic and elliptic nonlinear PDEs.
I should add that the abstract has many technical words, but the talk is quite easy to follow for anyone with a very basic knowledge of PDEs (and maybe even for those who don't have it...)
Feb 28 Mihalis Dafermos
University of Cambridge
The problem of stability for black hole spacetimes in general relativity
The notion of black hole plays a central role in general relativity. Nonetheless, the most basic mathematical questions about black holes remain unanswered, in particular, the question of their stability with respect to perturbation of initial data. In this talk, I will discuss how this problem is mathematically formulated, emphasizing its relation to decay properties for solutions of wave equations. I will then discuss recent progress on various related problems.
Mar 7 Jason Starr
SUNY Stony Brook and MIT
Rational simple connectedness and weak approximation
Rational connectedness and rational simple connectedness are algebraic analogues of path connectedness and simple connectedness obtained by replacing continuous maps from the unit interval with polynomial maps from the projective line. Given a system of polynomial equations in some variables and depending on one parameter, weak approximation is the problem of approximating to arbitrary order any power series solution in the parameter by a polynomial solution in the parameter. B. Hassett found a simple, elegant connection between rational simple connectedness and weak approximation. Recently A. J. de Jong and I proved smooth complete intersections of low degree are rationally simply connected, and thus they satisfy weak approximation.
Mar 14 Pavel Bleher
Indiana University Purdue University Indianapolis
Exact Solution of the Six-Vertex Model with Domain Wall Boundary Condition
The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $N$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an $N\times N$ Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large $N$ asymptotics of the six-vertex model with DWBC in the disordered phase. The solution is based on the Riemann-Hilbert approach and the Deift-Zhou nonlinear steepest descent method. As was noticed by Kuperberg, the problem of enumeration of alternating sign matrices (the ASM problem) is a special case of the the six-vertex model. We compare the obtained exact solution of the six-vertex model with known exact results for the 1, 2, and 3 enumerations of ASMs, and also with the exact solution on the so-called free fermion line. We prove the conjecture of Zinn-Justin that the partition function of the six-vertex model with DWBC has the asymptotics, $Z_N\sim CN^\kappa e^{N^2f}$ as $N\to\infty$, and we find the exact value of the exponent $\kappa$.
Mar 28

Apr 4 Edward Frenkel
UC Berkeley
Langlands Correspondence for Loop Groups
The local Langlands correspondence relates representations of a reductive algebraic group over the p-adic field and representations of the Galois group of this field. If we replace F by the field C((t)) of complex (formal) Laurent power series, then the corresponding group becomes the (formal) loop group. It is natural to ask: is there an analogue of the Langlands correspondence in this case? It turns out that the answer is affirmative, and there is an interesting theory which may be viewed as both "geometrization" and "categorification" of the classical theory. I will explain the general set-up for this new theory and give some examples using representations of affine Kac-Moody algebras. I will also explain the connection to the global geometric Langlands correspondence.
Apr 11 Manfred Einsiedler
Ohio State University
Spectral gap and effective equidistribution
The dynamics on homogeneous spaces has many interesting connections to number theory. One of the main problems here is to understand the distribution of closed orbits for subgroups H of the ambient Lie group G. In an ongoing joint work with G.Margulis and A.Venkatesh we prove an error rate in the equidistribution for semisimple subgroups H acting on congruence quotients of G. This makes use of spectral gap in the form of property (tau). However, the proof of our theorem can also be used to prove all cases of property (tau) except for groups of type A_1. We will discuss the relationship between spectral gap, effective decay of matrix coefficients, lattice counting, and effective equidistribution, as well as the main ideas of our argument.
Apr 18 Curtis T McMullen
Harvard University
Billiards and dynamics over moduli space
Apr 25 Jacob Rasmussen
Princeton University
Knot homologies
The notion of categorification replaces a polynomial invariant of knots - like the Alexander or Jones polynomial - with a family of graded vector spaces. Two main classes of these "knot homologies" are known: one springing from the work of Khovanov, and the other from the work of Ozsvath and Szabo. The parallels between the two types have some interesting geometric consequences, including a elementary proof of Milnor's conjecture on the slice genus of torus knots. I'll discuss some applications of these invariants to topology in dimensions 3 and 4 and describe some ideas about how the two different types should be related.
May 2 Tamar Ziegler
University of Michigan
Polynomial progressions in primes
In 1977 Szemeredi proved that any subset of the integers of positive density contains arbitrarily long arithmetic progression. A couple of years later Furstenberg gave an ergodic theoretic proof of Szemeredi's theorem. At around the same time Furstenberg and Sarkozy independently proved that any subset of the integers of positive density contains a perfect square difference, namely elements $x$, $y$ with $x-y=n^2$ for some positive integer n.
In 1995, Bergelson and Leibman proved, using ergodic theoretic methods, a vast generalization of both Szemeredi's theorem and the Furstenberg-Sarkozy theorem, establishing the existence of arbitrarily long polynomial progression in subsets of the integers of positive density.
The ergodic theoretic methods are limited, to this day, to handling sets of positive density. However, in 2004 Green and Tao proved that the question of finding arithmetic progressions in some special subsets of the integers of zero density - for example the prime numbers - can be reduced to that of finding arithmetic progressions in subsets of positive density. In recent work with T. Tao we show that one can make a similar reduction for polynomial progressions, thus establishing, through the Bergelson-Leibman theorem, the existence of arbitrarily long polynomial progressions in the prime numbers.

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