Global harmonic analysis and asymptotic geometry

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AbstractAward: DMS-0603850Principal Investigator: Steve ZelditchGlobal harmonic analysis is concerned with the impact of globalgeometry, particularly the geodesic flow, on the behavior ofeigenfunctions, eigenvalues and waves on a Riemannianmanifold. By comparison, local harmonic analysis uses only localanalysis on small balls. As an example of the impact of globalgeometry, ergodicity of the geodesic flow implies the uniformdistribution of eigenfunctions in the unit cotangent bundle. Anew result in this direction is part of the proposal: ergodicityof the geodesic flow can be used to determine the asymptotics ofthe complex nodal hypersurface of eigenfunctions. In anotherdirection, sup norms of eigenfunctions are related to existenceof focal points for the geodesic flow and to the first return mapon geodesic directions at the focal point (a joint project withC. Sogge and J. Toth). Asymptotic geometry aims to apply similarmethods to study problems in complex geometry, in particular tothe program of Yau, Tian, and Donaldson to study approximation ofall hermitian metrics on a positive line bundle by Bergmanmetrics. A joint project with J. Song is to use asymptotics ofBergman kernels on toric varieties to study a problem ofPhong-Sturm on how the geometry of the symmetric space of Bergmanmetrics of height N approaches the Monge-Ampere geodesic geometryof the full infinite dimensional space of hermitian metrics. Inanother direction, an ongoing joint project with B. Shiffman isto study statistical algebraic geometry of high degree varietiesas the degree tends to infinity. This has applications in stringtheory (joint work with M. Douglas).Both topics involve the application of ideas and methodsregarding the relation of classical and quantum mechanics togeometry and analysis. In each area there is a small "Planck'sconstant" which tends to zero. In the global analysis ofeigenfunctions, it is one over the eigenvalue; in geometry it isthe degree of a polynomial. As the "Planck constant" tends tozero, analytic objects (quantum mechanical) such aseigenfunctions or Bergman kernels tend to geometric objects(classical mechanics), which are much easier tounderstand. Physicists, engineers and mathematicians have beenstudying the relations between classical and quantum mechanicsfor almost a hundred years now, but the relations are sodifficult that fundamental problems remain. To take a venerableexample, 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 hownodal lines snake around on general drums. One project above isto determine the "complex" nodal line when billiards played onthe drum move in a chaotic way. Thus if one "thickens" the nodalline in complex directions, one can understand how it snakesaround on chaotic drum heads.