Mathematical theory of solids: From quantum mechanics to continuum models

The objective of this program is to understand the microscopic (atomistic, electronic) foundations of macroscopic (continuum) models of solids. This is a status report on the progresses that have been made and the challenges that remain. There are several motivations for such an initiative. One is intellectual: Similar program for uids (from kinetic theory or molecular dynamics to Euler or Navier-Stokes equations) has been a major driving force in mathematics, particularly in applied analysis. The second motivation is a more practical one: Solids exhibit a variety of physical properties (mechanical, thermal, electro-magnetic) and there exist a huge number of models, mostly ad hoc, for describing these properties. Some of these properties clearly have their origin at the microscopic level. For example, whether a solid is a metal or insulator is determined by the underlying electronic structure, and this in turn also affects the mechanical and thermal properties of the material. At the atomistic level, solid are crystal lattices. The lattice structure as well as the defects of the lattices directly inuence the macroscopic properties.