We will first introduce basic ideas of Compressive Sensing (CS), which
is an emerging
(some would say revolutionary) methodology in signal, image and data
processing.
The theory for CS has so far been built largely on a notion called
Restricted Isometry
Property or RIP. We will point out drawbacks of RIP-based analyses and
introduce
results from a non-RIP analysis including some new extensions. We will
also discuss
some related optimization algorithms.

A random walk is called recurrent if it is sure to return to its starting point and transient otherwise. A famous result of Polya is that simple symmetric random walk on the integer lattice is recurrent in dimensions 1 and 2, and transient in higher dimensions. We study random walks that are small perturbations of simple random walk. Our main result is that if the dimension is high enough then these random walks are transient.

We describe some recent developments involving particle methods in
electrostatics and fluid dynamics.

First we describe a Cartesian treecode for charged particle systems
undergoing screened Coulomb interactions in three dimensions. The
treecode reduces the operation count for computing potentials and
forces from $O(N^2)$ to $O(N\log N)$, where $N$ is the number of
particles in the system. The algorithm is especially well suited for
particles distributed on a surface. This is a step towards a treecode-
accelerated boundary integral solver for implicit solvent models in
biomolecular dynamics. (joint work with Peijun Li, Hans Johnston, and
Weihua Geng)

Second we describe a Lagrangian panel method for computing vortex
sheet motion in three dimensional fluid flow. Here too a treecode is
used, to evaluate the regularized Biot-Savart integral for the
sheet's self-induced velocity. The method is applied to compute the
azimuthal instability of a vortex ring. Details of the core dynamics
are clarified by tracking material lines on the sheet surface.
Results show the collapse of the vortex core in each wavelength and
the radial ejection of ringlets. These events are correlated with
local axial flow in the core of the ring. (joint work with Hualong
Feng and Leon Kaganovskiy)