We propose variational models for image inpainting in wavelet domain,
which aims to filling in missing or damaged wavelet coefficients
in image reconstruction. The problem is motaviated by error concealment
in image processing and communications. It is closely related to
classical image inpainting, with the difference being that the
inpainting regions are in the wavelet domain. This
brings new challenges to the reconstructions. The new variational
models, especially total variation minimization in conjunction
with wavelets lead to PDE's,in the wavelet domain and can be solved
numerically. The proposed models have effective and automatic control
over geometric features of the inpainted images including sharp edges, even
in the presence of substantial loss of wavelet coefficients, including in
the low frequencies. This work is joint with Tony Chan (UCLA) and
Jackie Shen (Minnesota).

What does a compact Riemannian manifold with unsolvable word problem look like from within? I will discuss Nabutovsky's work on the subject.

After introducing the concept of Maximal Regularity for parabolic problems and illustrating its usefulness for dealing with (fully) nonlinear problems, a brief introduction to Free Boundary Problems will be given. The focus will then shift to a class of Free Boundary Problems. Maximal regularity results as well as elliptic regularity results will be presented which are needed in the analysis of the Free Boundary Problems of interest.

Ordinary forms of weight at least 2 give rise to locally
reducible Galois representations. Greenberg has asked
whether these representations are semi-simple. One
expects this to be the case exactly when the underlying
form has CM. We shall speak about various results towards
this expectation that use p-adic families of forms
and deformation theory. This is joint work with Vatsal.

Numerical relativity has undergone a revolution during the past two years, with several groups now routinely performing accurate simulations of binary black hole systems with multiple orbits, mergers, and ringdown of the holes to a final single hole equilibrium state. This talk will discuss some of the mathematical developments that made this revolution possible. In particular new insights will be discussed into how the gauge degrees of freedom may be specified in the Einstein equations, and how this changes the behavior of the constraints of the theory.

The Robinson-Schensted correspondence is one between the
set of permutations and pairs of same-shape standard Young tableaux.
I'll recall a few of the combinatorial aspects of this.

The Steinberg scheme (for $GL_n$) is a set of triples, one nilpotent matrix
and two flags invariant under the nilpotent, whose components correspond
to permutations. I'll recall why this is (for those who haven't been
coming to the seminar), and show that they also correspond to pairs
of standard Young tableaux. The basic linkage between the linear algebra
and the combinatorics is that Jordan canonical forms of nilpotent matrices
correspond to partitions.

This talk will only require linear algebra, and a willingness to talk
about the ``components'' of an algebraic set.