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.

Consider a family of probability measures $\{\mu_y : y \in
\partial D\}$ on a bounded open domain $D\subset R^d$ with smooth
boundary.
For any starting point $x \in D$, we run a
a standard $d$-dimensional Brownian motion $B(t) \in R^d $ until it first
exits $D$ at time $\tau$,
at which time it jumps to a point in the domain $D$ according to the
measure $\mu_{B(\tau)}$ at the exit time,
and starts the Brownian motion afresh. The same evolution is repeated
independently each time the process reaches the boundary.
The resulting diffusion process is called Brownian motion with jump
boundary (BMJ).
The spectral gap of non-self-adjoint generator of BMJ, which describes the
exponential
rate of convergence to the invariant measure, is studied.
The main analytic tool is Fourier transforms with only real zeros.