Eigenfunctions in analysis, geometry and PDE

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I work on questions in spectral theory that have their origins in mathematical physics (spectra and eigenfunctions of Laplace-type and Dirac-type operators). I often study questions in the high-energy regime such as spectral statistics, as well as asymptotic properties of of high energy eigenstates: uniform distribution, $L^p$ norms, nodal and critical sets, semiclassical asymptotics. These questions lie at the intersection of analysis, partial differential equations, differential geometry, dynamical systems and mathematical physics. One of my main interests is the area of quantum chaos. The subject originated with the use of random matrix theory to model the spectra of atomic nuclei. Later it was recognized that similar phenomena occur in many other areas of mathematics and physics. I propose to continue my study of relationship between ergodic theory of partially hyperbolic dynamical systems, and high energy behavior of eigenfunctions of matrix-valued operators, such as Hodge Laplacian and Dirac operators. I also intend to study high energy spectra and eigenfunctions for systems where wave propagation involves branching (or ray-splitting); such systems arise, for example, in elasticity and acoustics. Many natural questions in semiclassical analysis of such systems remain unexplored, and require establishing new results in ergodic theory.