Global harmonic analysis and asymptotic geometry

Johns Hopkins University, Baltimore MD

Investigators

Abstract

Abstract Award: DMS-0603850 Principal Investigator: Steve Zelditch Global harmonic analysis is concerned with the impact of global geometry, particularly the geodesic flow, on the behavior of eigenfunctions, eigenvalues and waves on a Riemannian manifold. By comparison, local harmonic analysis uses only local analysis on small balls. As an example of the impact of global geometry, ergodicity of the geodesic flow implies the uniform distribution of eigenfunctions in the unit cotangent bundle. A new result in this direction is part of the proposal: ergodicity of the geodesic flow can be used to determine the asymptotics of the complex nodal hypersurface of eigenfunctions. In another direction, sup norms of eigenfunctions are related to existence of focal points for the geodesic flow and to the first return map on geodesic directions at the focal point (a joint project with C. Sogge and J. Toth). Asymptotic geometry aims to apply similar methods to study problems in complex geometry, in particular to the program of Yau, Tian, and Donaldson to study approximation of all hermitian metrics on a positive line bundle by Bergman metrics. A joint project with J. Song is to use asymptotics of Bergman kernels on toric varieties to study a problem of Phong-Sturm on how the geometry of the symmetric space of Bergman metrics of height N approaches the Monge-Ampere geodesic geometry of the full infinite dimensional space of hermitian metrics. In another direction, an ongoing joint project with B. Shiffman is to study statistical algebraic geometry of high degree varieties as the degree tends to infinity. This has applications in string theory (joint work with M. Douglas). Both topics involve the application of ideas and methods regarding the relation of classical and quantum mechanics to geometry and analysis. In each area there is a small "Planck's constant" which tends to zero. In the global analysis of eigenfunctions, it is one over the eigenvalue; in geometry it is the degree of a polynomial. As the "Planck constant" tends to zero, analytic objects (quantum mechanical) such as eigenfunctions or Bergman kernels tend to geometric objects (classical mechanics), which are much easier to understand. Physicists, engineers and mathematicians have been studying the relations between classical and quantum mechanics for almost a hundred years now, but the relations are so difficult that fundamental problems remain. To take a venerable example, Chladnyi first demonstrated two hundred years ago that, if one puts sand on a vibrating drum, the sand will move to the "nodal line". Despite the two hundred years, no one knows how nodal lines snake around on general drums. One project above is to determine the "complex" nodal line when billiards played on the drum move in a chaotic way. Thus if one "thickens" the nodal line in complex directions, one can understand how it snakes around on chaotic drum heads.