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.

The classical Friedrichs identity states that for $u\in C_0^{\infty}(\mathbb{R}^n)$ we have
$$\int_{\mathbb{R}^n} |D^2 u|^2\, dx = \int_{\mathbb{R}^n} |\Delta u|^2\, dx.$$
From this inequality we immediately get $W^{2,2}$-estimates for
solutions of $\Delta u =f$ and also for solutions of measurable perturbations
of the form $\sum_{ij}a_{ij}(x) u_{ij}(x)=f(x)$, when the matrix
$A=(a_{ij})$ is closed to the identity in sense made precise
by Cordes.
In this talk we first explore extensions of the Friedrichs identity in
the form of sharp inequalities
$$\int_{X} |\mathfrak{X}^2 u|^2\, dx \le C_1 \int_{X} |\Delta_{\mathfrak{X}} u|^2\, dx +C_2\int_{X} |\mathfrak{X}u|^2 \, dx $$
where $X$ is a Riemannian manifold, the Heisenberg group, and certain types of CR manifolds.
\par
We then show how to use these estimates to study quasilinear subelliptic equations.\par

This is joint work with Andr\`as Domokos (PAMS 133, 2005) and Sagun
Chanillo (2007 preprint.)

We proved that the integration of the Chern-Weil forms on CY moduli are always rational numbers. This result follows from a more general one: the integration of the Chern-Weil forms of the Hodge bundles on any coarse moduli spaces are rationa numbers. When the dimension of the moduli space is one, this was a result of Zucker and Peters. For the fisrt Chern class, this was proved by Kollar.

We will also discuss the applications in string theory. This is joint with M. Douglas.

A rotation number calculation for Jacobi marices with matrix entries
is presented. This allows to derive a formula for the density of states
in the case of a random Jacobi matrix with matrix entries. In order
to evaluate the appearing Birkhoff sums perturbatively with a good
control of the error terms, a certain Fokker-Planck operator on the
symmetric space of Lagrangian planes is used. The latter result
follows from a general pertubative analysis of random Lie group
actions on compact Riemannian manifolds.