This proposal is concerned with the asymptotics of eigenfunctions and eigenvalues of the Laplacian on Riemannian manifold, and analogues for the complex Laplacian on Kahler manifolds. Using methods of semi-classical analysis (the mathematics of the classical limit of quantum mechanics), the high frequency limit of eigenfunctions is related to the dynamics of the geodesic flow. The first application is to the well-known inverse spectral problem for analytic plane domains. The new idea is to use an exact formula for the Dirichlet resolvent and a Feynman diagram analysis of its trace. One obtains for the first time explicit formulae for so-called wave trace invariants in terms of the Taylor coefficients of the domain at endpoints of a bouncing ball orbit. From these one seeks to recover the Taylor coefficients and hence an analytic domain. So far the best result is that one can recover a mirror symmetric analytic domain from its spectrum. The second application is to geometry of eigenfunctions. Sogge and the proposer proved that the maximum possible growth rate of sup-norms of eigenfunctions only occurs on boundarlyless Riemannian manifolds which contain recurrent points, such that a positive measure of geodesics leaving the point return in a fixed time. One project is to prove this (if true) for bounded domains. A similar analysis is interesting for boundary values of eigenfunctions, which are important in boundary control theory. Hassell and the proposer have recently proved the ergodicity of boundary values of eigenfunctions when the billiard flow of the domain is ergodic, and this gave the first asymptotic estimates of even the L2 norms of the boundary values. Methods developed in that paper should have further applications to the geometry of eigenfunctions. The third project, joint with B. Shiffman is to apply semiclassical methods to statistical algebraic geometry, i.e., to finding statistical patterns in the zeros of polynomials. Among the patterns found so far are that zeros of systems of polynomials tend to repel in dimension one, be liike a neutral gas in dimension two and attract in higher dimensions. New problems are to explore the dependence of the distribution of zeros on the Newton polytope or on the number of monomials, as in Khovanski's fewnomial theory.
Semiclassical analysis is the area of mathematics which explores the classical limit of quantum mechancs, i.e.the limit where the fuzzy, jumpy small scale behavior of atoms and molecules merges with the solid, mechanical world we experience. Planck's constant h measures the length scale. Although it arose in physics, the same limit, as h tends to zero, arises in many problems in mathematics and science, often quite disconnected from the original physical background. This proposal is concerned with several problems of this kind. Several are traditional and well-known problems, for instance trying to determine a domain from its frequencies of vibration. Using semiclassical analysis of a kind more familiar to physicists than mathematicians (Balian-Bloch methods, Feynman diagrams), the first project is to obtain the best results to date on the problem, can one hear the shape of a drum. The results so far indicate that this method is better than any prior method. Another project is to view the degree of a polynomial as 1/h (inverse Planck constant) and to use semiclassical methods to find new statistical patterns in zeros and critical points of polynomials. One result is that the exponents which occur in the polynomials have a big impact on the positions of the zeros, leading to a tunnelling theory of zeros analogous to the tunnelling theory of electrons through barriers.
