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.

We consider only two fractals: Sierpinski and Apollonian gaskets. The
idea is to show on these two examples how geometry, analysis, algebra and
number theory are tied together in the simplest problems, related to
fractal sets.

We start with definitions, speculate on the general matrix numerical
systems, consider the analytic properties and the p-adic behavior of
harmonic functions, analyse the spectrum of the Laplace operator on the
Sierpinski gasket. Then we describe the geometry, group-theoretic
structure and arithmetic properties of the Apollonian gasket.

The final idea is to draw a parallel between the two fractals - an
unfinished program.

Gross and Pitaevskii proposed to model the dynamics of the Bose-Einstein
condensate by a nonlinear Schrdinger equation, the Gross-Pitaevskii
equation. This equation plays a key role in the theory and experiments of
the Bose-Einstein condensation. The fundamental mathematical question is
to derive this equation from the first principle physics law, the
many-body Schrdinger equation. In the time-independent setting, this
problem was solved by Lieb-Seiringer-Yngvason. In this lecture, we shall
review the recent progress concerning the dynamical aspects of this
problem and the analytic methods developed for quantum dynamics of
many-body systems.

The total variation based image denoising model of Rudin, Osher,
and Fatemi
has been generalized and modified in many ways in the literature; one of
these modifications is to use the L1 norm as the fidelity term. We study the
interesting consequences of this modification, especially from the point of
view of geometric properties of its solutions. It turns out to have
interesting
new implications for data driven scale selection and multiscale image
decomposition.

(joint work with Selim Esedgolu).