Geometric and dynamical aspects of spectral theory

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Geometry and dynamics on Riemannian manifolds are closely linked to the properties of eigenvalues and eigenfunctions of the Laplace operator. In this proposal I focus on two topics: dynamical aspects of spectral asymptotics and geometric inequalities for eigenvalues. The first topic is of particular importance from the quantum mechanical viewpoint. According to Bohr's correspondence principle, the spectral data at high energies ``feels'' the geodesic flow. My research program contributes to the mathematical understanding of this principle, particularly through the study of quantum ergodicity and of the asymptotic distribution of eigenvalues. The second topic is mostly concerned with the opposite - low - part of the spectrum. Geometric inequalities for small eigenvalues are of great importance in the study of sound propagation and heat diffusion, and they have attracted attention of mathematicians and physicists since the time of Lord Rayleigh. I discuss some ways to obtain new sharp eigenvalue inequalities on surfaces and planar domains, and to study the corresponding extremal geometries. I also indicate intriguing connections between this rather classical subject and spectral theory of the Dirac operator, which is a recently developed area of geometric analysis.