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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