Global harmonic analysis and asymptotic geometry

全局调和分析和渐近几何

• 批准号: 0603850
• 负责人: Steve Zelditch
• 金额: $ 26.7万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603850&HistoricalAwards=false
• 关键词: Global; harmonic; analysis; asymptotic; geometry

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.

AbstractAward：DMS-0603850首席研究员：Steve Zelditch全球调和分析关注全球几何的影响，特别是测地线流，对黎曼流形上的特征函数，特征值和波的行为。相比之下，局部谐波分析仅对小球进行局部分析。作为整体几何影响的一个例子，测地线流的遍历性意味着特征函数在单位余切丛中的均匀分布。在这个方向上的一个新的结果是建议的一部分：遍历的测地线流可以用来确定渐近的复杂节点超曲面的特征函数。 在另一个方向上，本征函数的超范数与测地流的焦点的存在以及焦点处测地方向上的第一个返回映射有关（与C. Sogge和J. Toth）。渐近几何旨在应用类似的方法来研究复几何中的问题，特别是Yau，Tian和唐纳森研究Bergman度量对正线丛上所有Hermitian度量的逼近的计划。一个联合项目与J.宋是使用渐近的伯格曼内核环面品种研究的问题Phong-Sturm如何对称空间的几何形状的伯格曼度量的高度N接近蒙日安培测地几何的全无限维空间的厄米度量。在另一个方向，一个正在进行的联合项目与B。Shiffman是研究当次数趋于无穷大时，高次变量的统计代数几何.这在弦理论中有应用（与M.道格拉斯）。这两个主题都涉及到关于经典力学和量子力学的关系、地理测量和分析的思想和方法的应用。在每一个区域都有一个小的“普朗克常数”趋于零。在整体分析中，它是特征值的1倍;在几何中，它是多项式的次数。当普朗克常数趋向于零时，分析对象（量子力学）如本征函数或伯格曼核趋向于几何对象（经典力学），这更容易理解。物理学家、工程师和数学家研究经典力学和量子力学之间的关系已经有近百年的历史了，但这些关系是如此困难，以至于基本问题仍然存在。举一个令人尊敬的例子，Chladnyi在200年前首次证明，如果把沙子放在振动的滚筒上，沙子会移动到“节点线”。 尽管有两百年的历史，没有人知道一般的鼓上的节点线是如何蜿蜒的。上面的一个项目是确定当台球在鼓上以混乱的方式运动时的“复杂”节点线。因此，如果一个人在复杂的方向上“加厚”了这条线，他就可以理解它是如何在混乱的鼓头上蜿蜒的。