I shall present an energetic variational phase field model for
multiphase incompressible flows which leads to a set of coupled
nonlinear system consisting a phase equation and the Navier-Stokes
equations. We shall pay particular attention to situations with large
density ratios as they lead to formidable challenges in both analysis
and simulation.

I shall present efficient and accurate numerical schemes for solving
this coupled nonlinear system, and show ample numerical results (air
bubble rising in water, Newtonian bubble rising in a polymeric fluid, defect
motion in a liquid crystal flow, etc.) which not only demonstrate the
effectiveness of the numerical schemes, but also validate the
flexibility and robustness of the phase-field model.

I will present some of the results by Marcy Barge and Jaroslaw Kwapisz (based on their paper "Geometric Theory of unimidular Pisot substitutions", Amer. J. Math., vol. 128 (2006), no. 5, pp. 1219--1282).

There are two classical ways of studying substitution tilings of the line: symbolic dynamics, and endomorphisms of ``train tracks". The authors give a strikingly new geometric approach and in particular show that if the tiling has a unimodular Pisot matrix of dimension d, then there is a factor onto the d-dimensional torus. In fact, they have a preprint removing the unimodular assumption. I propose to begin defining tilings and the tiling space X of a tiling T. X is a compact metric space that contains all tilings which have the same local patterns as T. In dimension 1 (the subject of this talk) X is similar to a solenoid.

I will not assume any familiarity with tiling theory.

Energy-driven pattern formation is difficult to define, but easy to recognize. I'll discuss two examples: (a) cross-tie wall patterns in magnetic thin films. (b) surface-energy-driven coarsening of two-phase mixtures. The two problems are rather different -- the first is static, the second dynamic. But they share certain features: in each case nature forms complex patterns as it attempts to minimize a suitable "free energy". The task of modeling and analyzing such patterns is a rich source of challenges -- many still open -- in the multidimensional calculus of variations.

This is a report on a joint work with Isabelle Catto, Norbert Mauser and Saber Trabelsi. The Multiconfiguration time dependent Hartree Fock Method (MCTDHF) is a nonlinear approximation of a linear system of /N/ quantum particles with binary interaction. It combines the principle of the Hartree Fock and the Galerkin approximation. The main difficulty is the introduction of a global (in space) density matrix $\Gamma(t) $ which may degenerate. By construction this approximation formally preserves the mass and the energy of the system. The conservation of energy can be used to balance the singularities Coulomb potential and to provide sufficient conditions for the global in time invertibility of $\Gamma(t)$.

In numerical computations this matrix is very often regularized (changed into $\Gamma(t) +\epsilon(t)$). In this situation the energy is no more conserved
and the mathematical analysis done in $L^2$ relies on Strichartz type estimates.