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).

From the point of view of mathematics,
micromagnetics is an ideal playground for a pattern forming system in materials
science: There are abundant experiments on a wealth of visually
attractive phenomena and there is a well--accepted continuum model.

In this talk, I will focus on a specific
experimental observation for thin film ferromagnetic elements: Elements
with elongated rectangular cross--section are saturated along the longer
axis by a strong external field. Then the external field is slowly
reduced. At a certain field strength, the uniform magnetization buckles
into a quasiperiodic domain pattern which resembles a concertina.
Our hypothesis is that the period of this pattern is the
frozen--in period of the unstable mode at critical field.

Starting point for the analysis is the micromagnetic model which has three
length scales. We rigorously identify four scaling regimes for the
critical field. One of the regimes has been
overseen by the physics literature. It displays an
oscillatory unstable mode, which we identify asymptotically.
In this parameter regime, we identify
a scaling limit for the bifurcation.

The analysis amounts to the combination of an asymptotic
limit with a bifurcation argument. This is carried out by a suitable
``blow--up'' of the energy landscape in form of Gamma--convergence.
Numerical simulation of the normal form visualizes a homotopy
from a turning point to
the strongly nonlinear concertina pattern.
This is joint work with Ruben Cantero--Alvarez and Jutta Steiner.