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.
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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.