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.
 

Efficient representation of geometric models is essential in many applications in Geometric Design and Computer Graphics. In this talk, I will discuss two general approaches for efficient representations of geometric models--adaptive representation and sparse representation. For adaptive representation, we propose splines over T-meshes. We discuss some theoretic issues such as dimension calculation and basis constructions of the spline spaces. We then apply splines over T-meshes in adaptive modeling of geometric models. For sparse representation, we apply the L_1 optimization technique in surface denoising and 3D printing.  Further research problems are also discussed

We consider the Williams Bjerknes model, also known as the biased voter model on the d-regular tree T^d, where d \geq 3. Starting from an initial configuration of ``healthy'' and ``infected'' vertices, infected vertices infect their neighbors at Poisson rate \lambda \geq 1, while healthy vertices heal their neighbors at Poisson rate 1. All vertices act independently. It is well known that starting from a configuration with a positive but finite number of infected vertices, infected vertices will continue to exist at all time with positive probability iff \lambda > 1. We show that there exists a threshold \lambda_c \in (1, \infty) such that if \lambda > \lambda_c then in the above setting with positive probability all vertices will become eventually infected forever, while if \lambda < \lambda_c, all vertices will become eventually healthy with probability 1. In particular, this yields a complete convergence theorem for the model and its dual, a certain branching coalescing random walk on T^d -- above \lambda_c. We also treat the case of initial configurations chosen according to a distribution which is invariant or ergodic with respect to the group of automorphisms of T^d. Joint work with A. Vandenberg-Rodes, R. Tessler.

The notion of an orthogonal polynomial ensemble generalizes many
important point processes arising in random matrix theory, probability
and combinatorics.
This talk describes recent joint work with Maurice Duits dealing with
the fluctuations of the random empirical measure for general
orthogonal polynomial ensembles, on all scales, for both varying and
fixed measures.
We obtain a general concentration inequality and prove both global
(`macroscopic') and local (`mesoscopic') almost sure convergence of
linear statistics under fairly weak assumptions on the ensemble. An
important role in the analysis is played by a strengthening of the
Nevai condition from the theory of orthogonal polynomials.
No previous knowledge of orthogonal polynomial ensembles or orthogonal
polynomial theory is assumed.