Asymptotic Geometry of Eigenfunctions and Polynomials

本征函数和多项式的渐近几何结构：这一建议涉及黎曼流形上拉普拉斯函数的本征函数和本征值的渐近性质，以及Kahler流形上复拉普拉斯函数的类似问题。利用半经典分析方法(量子力学经典极限的数学)，将本征函数的高频极限与测地线流的动力学联系起来。第一种方法应用于众所周知的解析平面域的谱反问题，新的思想是使用Dirichlet预解器的精确公式及其迹的Feynman图分析。首次得到了所谓的波迹不变量的显式公式，它是根据弹跳球轨道的区域端点的泰勒系数来表示的。从中求出泰勒系数，从而求出一个解析域。到目前为止，最好的结果是可以从其频谱中恢复镜像对称解析区域。第二个应用是本征函数的几何。Sogge和作者证明了特征函数的超范数的最大可能增长率只出现在含有回归点的无边界黎曼流形上，即测地线在固定时间离开点返回的正度量。一个项目是为有界域证明这一点(如果是真的)。特征函数的边值问题也有类似的分析，这在边界控制理论中是很重要的。最近，Hassell和作者证明了当区域的弹流是遍历时，特征函数边值的遍历性，并给出了边值的L2范数的第一个渐近估计。文中提出的方法应进一步应用于特征函数的几何。第三个项目，与B.Shiffman合作，将半经典方法应用于统计代数几何，即在多项式的零点中寻找统计模式。到目前为止发现的模式中，多项式系统的零点在一维中往往相互排斥，在二维中像中性气体一样，在更高的维中吸引。新的问题是探索零点分布对牛顿多面体或单项式个数的依赖，如霍万斯基的少数正统理论。半经典分析是探索量子力学的经典局限性的数学领域，即原子和分子的模糊、跳跃的小尺度行为与固体机械世界的经验相融合的极限。普朗克常数h测量长度尺度。虽然它出现在物理学中，但同样的极限，当h趋于零时，在数学和科学中的许多问题中都会出现，往往与最初的物理背景完全脱节。这项建议涉及到几个这类问题。有几个是传统的和众所周知的问题，例如试图从振动频率确定一个域。使用一种比数学家更熟悉的主题物理学家的半经典分析(巴利安-布洛赫方法，费曼图)，第一个项目是在这个问题上获得迄今为止最好的结果，人们可以听到鼓的形状。到目前为止的结果表明，该方法优于以往的任何方法。另一个项目是考虑抛物线式的次数1/h(逆普朗克常数)，并使用半经典方法在多项式的零点和临界点找到新的统计模式。一个结果是，出现在多项式中的指数对零点的位置有很大的影响，导致了零点隧穿理论，类似于电子通过势垒的隧穿理论。

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.