Progress in the mathematical theory of quantum disordered systems

We review recent progress in the mathematical theory of quantum disordered systems: the Anderson transition (joint work with Domingos Marchetti), the (quantum and classical) Edwards-Anderson (EA) spin glass model and return to equilibrium for a class of spin glass models, which includes the EA model initially in a very large transverse magnetic field.Comment: 25 pages, 1 figure, based on lectures at Bressanone and G\"ottinge

A precise formulation of the third law of thermodynamics

The third law of thermodynamics is formulated precisely: all points of the state space of zero temperature $\Gamma_0$ are physically adiabatically inaccessible from the state space of a simple system. In addition to implying the unattainability of absolute zero in finite time (or "by a finite number of operations"), it admits as corollary, under a continuity assumption, that all points of $\Gamma_0$ are adiabatically equivalent. We argue that the third law is universally valid for all macroscopic systems which obey the laws of quantum mechanics and/or quantum field theory. We also briefly discuss why a precise formulation of the third law for black holes remains an open problem.Comment: 24 pages, no figure