Conventional wisdom and common practice in acquisition and
reconstruction of images from frequency data follows the basic
principle of the Nyquist density sampling theory. This principle
states that to reconstruct an image, the number of Fourier samples we
need to acquire must match the desired resolution of the image, i.e.
the number of pixels in the image.

This talk introduces a newly emerged sampling theory which shows that
this conventional wisdom is inaccurate. We show that perhaps
surprisingly, images or signals of scientific interest can be
recovered accurately and sometimes even exactly from a limited number
of nonadaptive random measurements. In effect, the talk introduces a
theory suggesting "the possibility of compressed data acquisition
protocols which perform as if it were possible to directly acquire
just the important information about the image of interest." In other
words, by collecting a comparably small number of measurements rather
than pixel values, one could in principle reconstruct an image with
essentially the same resolution as that one would obtain by measuring
all the pixels, a phenomenon with far reaching implications.

The reconstruction algorithms are very concrete, stable (in the sense
that they degrade smoothly as the noise level increases) and
practical; in fact, they only involve solving convenient convex
optimization programs. If time allows, I will discuss connections
with other fields such as statistics and coding theory.

Since the discovery in 1993 of the figure-8 orbit by Cris Moore, a large number of periodic orbits for equal n masses have been found having beautiful symmetries and topologies. Most of these orbits are either planar or have been obtained from perturbation of planar orbits.

Recently Moore and I have found also a number of new three-dimensional periodic orbits of this kind which have cubic symmetries. We found these orbits by symmetry considerations, and by minimizing numerically the action integral directly as a function of the Fourier coefficients for the periodic orbit coordinates. I will review some of the early history of periodic orbits, discuss our method, and present video animations of recent results.

Oscillations and other patterns of neuronal activity arise throughout
the central nervous system. This activity has been observed in sensory
processing, motor activities, and learning, and has been implicated in
the generation of sleep rhythms, epilepsy, and parkinsonian tremor.
Mathematical models for neuronal activity often display an incredibly
rich structure of dynamic behavior. In this lecture, I describe how the
neuronal systems can be modeled, various types of activity patterns that
arise in these models, and mechanisms for how the activity patterns are
generated. In particular, I demonstrate how methods from geometric
singular perturbation theory have been used to analyze a recent model
for activity patterns in an insect's antennal lobe.

The digital nature of biology is crucial to its functioning
as an information system. The hierarchical development of
biological components (translating DNA to proteins which form
complexes in cells that aggregate to make tissue which form
organs in different species) is discrete (or quantized) at
each step. It is important to understand what makes proteins
bind to other proteins predictably and not in a continuous
distribution of places, the way grease forms into blobs.

Data mining is a major technique in bioinformatics. It has been
used on both genomic and proteomic data bases with significant
success. One key issue in data mining is the type of lens that
is used to examine the data. At the simplest level, one can just
view the data as sequences of letters in some alphabet. However,
it is also possible to view the data in a more sophisticated
way using concepts and tools from physical chemistry. We will
give illustrations of the latter and also show how data mining
(in the PDB) has been used to derive new results in physical
chemistry. Thus there is a useful two-way interaction between
data mining and physical chemistry.

We will give a detailed description of how data mining in the
PDB can give clues to how proteins interact. This work makes
precise the notion of hydrophobic interaction in certain cases.
It provides an understanding of how molecular recognition and
signaling can evolve. This work also introduces a new model of
electrostatics for protein-solvent systems that presents
significant computational challenges.

We study the regularity for the tangential Cauchy-Riemann equations and the associated Laplacian on CR manifolds with minimal smoothness assumption. One application is to extend the embedding theorem of Boutet De Monvel
to strongly pseudoconvex CR manifolds of class C^2.

(Joint work with Lihe Wang).