For closed surfaces and for surfaces with boundary there are natural eigenvalue extremal problems whose solutions, when they exist, determine minimal surfaces in the sphere or the ball with a natural boundary condition. We will discuss the existence problem and describe some geometric properties of extremal metrics.

Consider a Brownian particle in a prescribed time-intependent incompressible flow in a bounded domain. We investigate how the strength of the flow and its geometric properties affect the expected exit time of the particle. The two main questions we analyze in this talk are as follows.  1. Incompressible flows are known to enhance mixing in many contexts, but do they also always decrease the exit time? We prove that the answer is no, unless the domain is a disk. 2.  Suppose the flow is cellular with amplitude A, and the domain is of size L.   What could be said about the exit time when both L and A are large? We prove that there are two characteristic regimes: a) if L << A^4, then the exit time from the entire domain is compatible with the exit time from a single flow cell, and it can be determined from the Freidlin–Wentzell theory; b) if L>> A^4, then the problem `homogenizes' and the exit time is determined by the effective diffusivity of cellular flows.

A Hodge class on a smooth complex projective variety gives rise to an associated hermitian line bundle on a Zariski open subset of a complex projective space P^n.  I will discuss recent work with P. Brosnan which shows that the Hodge conjecture is equivalent to the existence of a particular kind of degenerate behavior of this metric near the boundary.

 We study the path properties of the random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense ``tight in probability'' as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer.

Highly coherent sensing matrices arise  in discretization of continuum problems such as radar  and medical imaging when the grid spacing is below the Rayleigh threshold as well as in using highly coherent, redundant dictionaries as sparsifying operators.

We propose new algorithms (BLOOMP, BP-BLOT)  based on techniques of band exclusion and local optimization to enhance existing compressed sensing algorithms (OMP, BP) and deal with such coherent sensing matrices.

 BLOOMP has provably performance guarantee of reconstructing sparse, widely separated objects independent  of the redundancy and have a sparsity constraint and computational
cost similar to OMP's.

We demonstrate the effectiveness of our schemes in various compressed sensing  problems with highly coherent, redundant sensing matrices.