There are no analogs of Navier-Stokes equations for solid mechanics. One reason is that information at the atomic scale seems to play a much more important role for solids than for fluids. A satisfactory mathematical theory for solids has to taken into account the behavior of solids at different scales, from electronic to atomic, to macroscopic scales.

I will discuss some of the fundamental problems that we have to resolve in order to build such a theory. I will start by reviewing the geometry of crystal lattices, the quantum as well as classical atomistic models of solids. I will then focus on a few selected problems:

(1) The crystallization problem -- why the ground states of solids are crystals and which crystal structure do they select?

(2) the microscopic foundation of elasticity theory;

(3) stability and instability of crystals;

(4) the electronic structure and density functional theory.