The only known sufficient condition for the existence of a Kahler-Einstein metric on a Fano manifold can be formulated in terms of so-called alpha-invariant introduced by Tian and Yau more than 20 years ago. This invariant can be naturally defined for log Fano varieties with log terminal singularities using purely algebraic language. Using a global-to-local results of Shokurov, one can define a similar invariants for a germ of log terminal singularity. We describe the role played by alpha-invariants in birational geometry and singularity theory. We prove the existence of Kahler-Einstein metrics on many quasismooth well-formed weighted del Pezzo hypersurface and compare this result with new obstructions found by J.Gauntlett, D.Martelli, J.Sparks and S.-T.Yau. We apply our technique to classify weakly-exceptional quasismooth well-formed weighted del Pezzo hypersurface using the classification of isolated rational quasihomogeneous three-dimensional singularities obtained by S.S.T.Yau and Y.Yu.

In this talk, asymptotic limit problems of compressible Euler-Maxwell
equations in plasma physics are discussed. Some recent results about the rigorous convergence of compressible Euler-Maxwell systems to the incompressible Euler or e-MHD equations are given and some new methods or ideas are reviewed.

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).