I will explain how the so-called "infinity Laplacian'' PDE arrises in sup-norm variational problems and discuss the regularity of solutions.

One-Frequency Schrödinger operators give one of the simplest models where fast transport and localization phenomena are possible. From a dynamical perspective, they can be studied in terms of certain one-parameter families of quasi-periodic co-cycles, which are similarly distinguished as simplest classes of dynamical systems compatible with both KAM phenomena and non-uniform hyperbolicity (NUH). While much studied since the 1970's, until recently the analysis was mostly confined to ''local theories'' describing the KAM and the NUH regimes in detail. In this talk we will describe some of the main aspects of the global theory that has been developed in the last few years.

An important question is to non-invasively find the volume of each phase in body, by only probing its response at the boundary. Here we consider a body containing two phases arranged in any configuration, and address the inverse problem of bounding the volume fraction of each phase from electrical tomography measurements at the boundary, i.e. measurements of the current flux through the boundary produced by potentials applied at the boundary. It turns out that this problem is closely related to the extensively studied problem of bounding the effective conductivity of periodic composite materials. Those bounds can be used to bound the response of an arbitrarily shaped body, and if this response has been measured, they can be used to extract information about the volume fraction.

Numerical experiments show that for a wide range of inclusion shapes one of the bounds turns out to be close to the actual volume fraction. The bounds extend those obtained by Capdeboscq and Vogelius for asymptotically small inclusions. The same ideas can be extended to elasticity and used to incorporate thermal measurements as well as electrical measurements. The translation method for obtaining bounds on the effective conductivity can also be applied directly to bound the volume fraction of inclusions in a body. This is joint work with Hyeonbae Kang and Eunjoo Kim.

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.