The state-of-art methods for simulating high frequency wave propagation rely on the high frequency theories, i.e., geometrical optics.  The waves  before caustics appear can be constructed accurately by geometrical optics if source singularities for the amplitude and phase can be resolved.  In this talk, we introduce systematic approaches to resolve the source singularities of amplitude and phase. More importantly, we introduce a new method  to simulate
the highly oscillating waves beyond geometrical optics even when caustics appear. Numerical examples will be presented to demonstrate the methods. (with Profs. Robert Burridge, Jianliang Qian and Hongkai Zhao).

What is the probability that a random abelian variety over F_q is ordinary? Using (semi)linear algebra, we will answer an analogue of this question, and explain how our method can be used to answer similar statistical questions about p-rank and a-number. The answers are perhaps surprising, and deviate from what one might expect via naive reasoning. Using these computations and numerical evidence, we formulate several ``Cohen-Lenstra" heuristics for the structure of the p-torsion on the Jacobian of a random hyperelliptic curve over F_q. These heuristics are the "l=p " analogue of Cohen-Lenstra in the function field setting. This is joint work with Jordan Ellenberg and David Zureick-Brown.

To a Hilbert modular form one may attach a p-adic analytic
L-function interpolating certain special values of the usual L-function.
Conjectures in the style of Mazur, Tate and Teitelbaum prescribe the order
of vanishing and first Taylor coefficient of such p-adic L-functions, the
first coefficient being controlled by an L-invariant which has conjectural
(arithmetic) value defined by Greenberg and Benois. I will explain how to
compute arithmetic L-invariants for (critical, exceptional) symmetric
powers of non-CM Iwahori level Hilbert modular forms via triangulations on
eigenvarieties. This is based on joint work with Robert Harron.

There is a good deal of resemblence of CR geometry in dimension three with conformal geometry
in dimension four. Exploiting this resemblence is quite fruitful. For instance, the presence of
several conformally covariant operators in both geometries allows us to formulate correct
conditions for the embedding problem as well as the CR Yamabe problem. There is also
large difference in the presence of pluriharmonic functions. I will also describe a new
operator which gives control of the pluriharmonics and allows a formulation of a
sphere theorem in this geometry.