Wednesday,
November 5, 2008
4:15
p.m. // 151 Sloan
Monica Visan (University of Chicago)
Nonlinear Schrodinger equations at critical regularity.
Abstract: We introduce the nonlinear Schrodinger equation
(NLS) and define criticality.
We then survey the history of the two most studied cases of critical
NLS, namely, the masscritical NLS and the energycritical NLS. This
includes recent joint work with Rowan Killip, Terry Tao and Xiaoyi
Zhang.
Thursday,
November 6, 2008
4:15
p.m. // 151 Sloan
Rowan Killip (UCLA)
Some
operators with randommatrix eigenvalue statistics
Abstract: Several deterministic (i.e. manifestly
nonrandom) sequences have been observed numerically to exhibit the
same local behaviour as the eigenvalues of a random matrix. One example
is characteristic frequencies of certain oddly shaped drums.
In this talk I will describe some individual operators (of less
intrinsic interest) which have the same local eigenvalue statistics.
Much of the talk will be devoted to the journey leading to this result,
including the suprising role of numerical linear algebra.
Tuesday,
November 11, 2008
4:15
p.m. // 151 Sloan
Sourav Chatterjee (Berkeley)
A
rigorous theory of chaos in disordered systems
Abstract: Disordered systems are an important class of
models in statistical mechanics, having the defining characteristic
that the energy landscape is a fixed realization of a random field.
Examples include various models of glasses and polymers. They also
arise in other subjects, like fitness models in evolutionary biology.
The ground state of a disordered system is the state with minimum
energy. The system is said to be chaotic if a small perturbation of the
energy landscape causes a drastic shift of the ground state. In this
talk I will present a rigorous theory of chaos in disordered systems
that confirms longstanding physics intuition about connections between
chaos, anomalous fluctuations of the ground state energy, and the
existence of multiple valleys in the energy landscape. Combining these
results with mathematical tools like hypercontractivity, I will present
a proof of the existence of chaos in directed polymers. This is the
first rigorous proof of chaos in any nontrivial disordered
system. Applications to other models like spin glasses, fitness models,
and general Gaussian fields will also be discussed.
Tuesday,
November 18, 2008
4:15
p.m. // 151 Sloan
Yi Ni (MIT)
Floer
homology and fibered 3manifolds
Abstract: There are several Floer homologies of
3manifolds defined using gauge theory and symplectic geometry. Three
of them are Instanton Floer homology, Monopole Floer homology and
Heegaard Floer homology. It turns out that each of these three
homologies determines whether a 3manifold is a surface bundle over the
circle. I will give a survey on this topic,
which will cover the works of Ghiggini, Ni, Juhasz, Kronheimer and
Mrowka.
Tuesday, November 25, 2008
4:15
p.m. // 151 Sloan
Adrian Ioana (Caltech and CMI)
Rigidity in orbit equivalence and von Neumann algebras.
Abstract: The first examples of von Neumann algebras came
from actions of countable groups on probability spaces. As it turns
out, their study (up to
isomorphism) is closely related to the study of countable equivalence
relations (up to orbit equivalence). Recently, this connection proved
to be very useful, leading to remarkable rigidity results in both orbit
equivalence and von Neumann algebras theory. In this talk, I will
survey some of these results including: existence of II_1 factors
without symmetries, new cocycle superrigidity results, and existence of
nonorbit equivalent actions for arbitrary nonamenable groups.
Tuesday,
February 17, 2009
4:15
p.m. // 151 Sloan
Zeev Dvir (Institute
for Advanced Study)
The finite field Kakeya problem
Abstract: The finite field
Kakeya problem deals with finding lower bounds on the size of sets in a
vector space over a finite field that contain a line in every
direction. This problem was introduced by Wolff in 1999 and is
connected to several problems in analysis, combinatorics and
theoretical computer science.
In this talk I will survey recent progress on this problem [Dvir 08,
Saraf Sudan 08, Dvir Kopparty Saraf Sudan 09] which give nearly optimal
bounds on the size of Kakeya sets.
