Computation of high frequency solutions to wave equations is important
in many applications, and notoriously difficult in resolving wave
oscillations. Gaussian beams are asymptotically valid high frequency solutions
concentrated on a single curve through the physical domain, and superposition
of Gaussian beams provides a powerful tool to generate more general high
frequency solutions to PDEs. In this talk I will present a recovery theory of
high frequency wave fields from phase space based measurements. The
construction use essentially the idea of Gaussian beams, level set description
in phase space as well as the geometric optics. Our main result asserts that
the kth order phase space based Gaussian beam superposition converges to the
original wave field in L2 at the rate of $\epsilon^{k/2-n/4} in dimension $n$.
The damage done by caustics is accurately quantified. Though some calculations
are carried out only for linear Schroedinger equations, our results and
main arguments apply to more general linear wave equations. This work is in
collaboration with James Ralston (UCLA).

The Hodge group (aka special Mumford-Tate group) of a complex abelian variety $X$ is a certain linear reductive algebraic group over the rationals that is closely related to the endomorphism ring of $X$. (For example, the Hodge group is commutative if and only if $X$ is an abelian variety of CM-type.) In this talk I discuss" lower bounds" for the center of Hodge groups of superelliptic jacobians. (This is a joint work with Jiangwei Xue.)