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.

The least action principle of Fermat, Maupertuis, Lagrange, Hamilton
and others lies at the foundation of optics and classical mechanics. Given the
initial and final positions of a system of particles (rays), the orbit of the
particles (rays) is determined by minimizing the action. I shall describe a
generalization of this principle that applies to waves. The principle will be
derived first for the Schroedinger equation, and then it will be generalized
to other wave equations and to singular solutions. I shall
also consider the application of the principle to the design of phase sensors
and illumination systems.

I will discuss energetics and regularity for systems
of equations describing fluids with microscopic inclusions.

Thirty million Americans suffer from hearing loss. Several generations of the cochlear implant have been developed to help restore hearing to people with severe-to- profound hearing loss. The cochlear implant is surgically implanted in the inner ear and stimulates the auditory nerve with electric currents that encode certain acoustic features. We will review speech processing strategies that have been used for cochlear implants and discuss their performance and limitations. In general, the present cochlear implant users can understand 70-80% speech in quiet, allowing them to communicate over the telephone. However, their speech performance drops dramatically in a noisy environment. In addition, they cannot fully appreciate music. We will present a new sound processing strategy, which can dynamically encode the slowing varying as well as fast varying components in speech and music sounds.
Several experiments were performed to establish the utilities and limitations of the new speech strategy. A frequency modulation detection experiment revealed that cochlear implant users could only discern dynamic rate changes at low frequencies. Novel pulse trains, mimicking the stochastic neuron-firing pattern in normal acoustic hearing, were employed to improve rate pitch perception in cochlear implant users. Electrode ranking and pitch estimation experiments were conducted to examine the relative contributions of rate and place cues to pitch perception in electric hearing. Finally, the novel speech strategy was implemented on a real-time DSP processor, in which pitch information is encoded by varying the stimulation rate and the site of the stimulation. Results showed that optimally combining the rate and place pitch cues can improve the current cochlear-implant users melody recognition by as much as 36%. The new strategy has a great potential to improve cochlear implant users'