A typical step in matrix algebra is elimination, and its description as a triangular factorization. For a doubly infinite banded Toeplitz matrix A, that step is made easy by factoring the polynomial a(z) whose coefficients come from the diagonals of A. What to do if A is not Toeplitz?

A nice case is a permutation matrix (on Z). Which is the main diagonal? For the (Toeplitz) example of a shift matrix, the main diagonal contains the 1's. We identify the correct diagonal for every banded permutation. Then we consider banded matrices (not Toeplitz!) as operators on L2(Z) and ask about their factorization.

A special case is when the inverse of A is also banded -- these matrices factor into block-diagonal matrices. The help coming from analysis is the theory of Fredholm operators.

One can rarely identify extremizers of nontrivial inequalities, yet one can not infrequently show that extremizers exist. Thus one asks what qualitative and quantitative properties can be established. One way to attack such questions is to exploit Euler-Lagrange equations which extremizers must satisfy. Inequalities involving L^p norms, with p not equal to 2, lead to nonlinear equations. In this talk we discuss the nonlinear, nonlocal Euler-Lagrange equation which arises in connection with such an inequality for the Radon transform. We show that all solutions are infinitely differentiable, and have a certain rate of decay at spatial infinity. (joint work with Qingying Xue)

TBD

This is joint work with Louis-Pierre Arguin, Michael Damron and Dan Stein (arXiv:0911.4201). It is an open problem to determine the number of infinite-volume ground states in the Edwards-Anderson (nearest neighbor) spin glass modelon Z^d for d \geq 2 (with, say, mean zero Gaussian couplings). This is a limiting case of the problem of determining the number of extremal Gibbs states at low temperature. In both cases, there are competing conjectures for d \geq 3, but no complete results even for d=2. I report on new results which go some way toward proving that (with zero external field, so that ground states come in pairs, related by a global spin flip) there is only a single ground state pair (GSP). Our result is weaker in two ways: First, it applies not to the full plane Z^2, but to a half-plane. Second, rather than showing that a.s. (with respect to the quenched random coupling realization J) there is a single GSP, we show that there is a natural joint distribution on J and GSP's such that for a.e. J, the conditional distribution on GSP's given J is supported on only a single GSP. The methods used are a combination of percolation-like geometric arguments with translation invariance (in one of the two coordinate directions of the half-plane) and uses as a main tool the "excitation metastate" which is a probability measure on GSP's and on how they change as one or more individual couplings vary.

Bregman iteration has been around since 1967. It turns out to be unreasonably effective for optimization problems involving L1, BV and related penalty terms. This is partly because of a miraculous cancellation of error. We will discuss this and give biomedical imaging applications, related to compressive sensing and Total Variation based restoration.