Asymptotic Geometry of Eigenfunctions and Polynomials

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ASYMPTOTIC GEOMETRY OF EIGENFUNCTIONS AND POLYNOMIALSAbstract:This proposal is concerned with the asymptotics of eigenfunctions and eigenvalues of the Laplacian on Riemannian manifold, and analoguesfor the complex Laplacian on Kahler manifolds. Using methods ofsemi-classical analysis (the mathematics of the classical limit ofquantum mechanics), the high frequency limit of eigenfunctions is relatedto the dynamics of the geodesic flow. The first application is to thewell-known inverse spectral problem for analytic plane domains.The new idea is to use an exactformula for the Dirichlet resolvent and a Feynmandiagram analysis of its trace. One obtains for the first time explicit formulae for so-calledwave trace invariants in terms of the Taylor coefficients of the domain atendpoints of a bouncing ball orbit. From these one seeks to recover theTaylor coefficients and hence an analytic domain. So far the best resultis that one can recover a mirror symmetric analytic domain from itsspectrum. The second applicationis to geometry of eigenfunctions. Sogge and the proposer proved that the maximum possible growth rate of sup-norms of eigenfunctions onlyoccurs on boundarlylessRiemannian manifolds which contain recurrent points, such thata positive measure of geodesics leaving the point return in a fixedtime. One project is to prove this (if true) for bounded domains. A similar analysis is interesting for boundary values of eigenfunctions, whichare importantin boundary control theory. Hassell and the proposer have recentlyprovedthe ergodicity of boundary values of eigenfunctions when the billiardflowof the domain is ergodic, and this gave the first asymptotic estimatesofeven the L2 norms of the boundary values. Methods developed in thatpapershould have further applications to the geometry of eigenfunctions. Thethird project,joint with B. Shiffmanis to apply semiclassical methods to statistical algebraic geometry,i.e., tofinding statistical patterns in the zeros of polynomials. Among thepatterns foundso far are that zeros of systems of polynomials tend to repel indimension one,be liike a neutral gas in dimension two and attract in higherdimensions. Newproblems are to explore the dependence of the distribution of zeros onthe Newtonpolytope or on the number of monomials, as in Khovanski's fewnomialtheory.Semiclassical analysis is the area of mathematics which explores theclassicallimit of quantum mechancs, i.e.the limit where the fuzzy, jumpy smallscale behaviorof atoms and molecules merges with the solid, mechanical world weexperience.Planck's constant h measures the length scale. Although it arose inphysics,the same limit, as h tends to zero, arises in many problems inmathematics andscience, often quite disconnected from the original physicalbackground. Thisproposal is concerned with several problems of this kind. Several aretraditionaland well-known problems, for instance trying to determine a domain fromitsfrequencies of vibration. Using semiclassical analysis of a kind morefamiliar tophysicists than mathematicians (Balian-Bloch methods, Feynmandiagrams), the first project is to obtain the best results to date on the problem,can onehear the shape of a drum. The results so far indicate that this methodis betterthan any prior method. Another project is to view the degree of apolynomialas 1/h (inverse Planck constant) and to use semiclassical methods tofind newstatistical patterns in zeros and critical points of polynomials. Oneresult isthat the exponents which occur in the polynomials have a big impact onthepositions of the zeros, leading to a tunnelling theory of zerosanalogousto the tunnelling theory of electrons through barriers.