Mini-Courses

Pablo A Ferrari - Invariant measures for particle systems and automata

The open boundary Kipnis-Presutti-Marchioro particle model on a finite graph is a symmetric evolution producing mass flow when the boundary conditions are distinct. The continuous Box-Ball-System is a one-dimensional transport automaton showing soliton behavior. We will construct invariant measures for those systems. The KMP invariant measure is a product of exponential random variables with random means, called "hidden temperature''. The soliton decomposition of a BBS random configuration is a point process, translated hierarchically under the BBS induced dynamics. These properties are used to construct a huge family of invariant measures.

Michael Loss - Sharp Spinor Inequalities and Zero Modes of the Dirac Equation

For atoms in magnetic fields the spinor structure of the electrons is important. It is this fact that causes certain materials to be paramagnetic. Almost 40 years ago it was realized that the spinor structure also plays a peculiar role for the stability of matter in the form of zero modes for the three dimensional Dirac equation. As a side remark zero modes also play a destructive role in the fermion problem in Quantum Electrodynamics. It is a reasonable goal to ask for sharp conditions on the magnetic field guarantee the absence of such zero modes. This leads to consider spinor inequalities and I try to explain a few approaches and formulate open questions. This is based on work with Rupert Frank.

Topics: 1) Atoms in magnetic fields, diamagnetic inequality, paramagnetism, stability of atoms.

             2) Zero modes and the Hopf fields

             3) Regularity of zero modes

             3) Sobolev's inequality and a first criterion for absence of zero modes

             4) Some sharp spinor inequalities

             5) An improved diamagnetic inequality

             6) Are the Hopf fields optimal for zero modes? The Schrödinger-Lichnerowicz identity