Global harmonic analysis and asymptotic geometry

摘要奖：DMS-0603850首席研究员：史蒂夫·泽尔迪奇全球调和分析关注全球几何，特别是测地线流对黎曼流形上特征函数、特征值和波的行为的影响。相比之下，局部谐波分析只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本征函数在单位余切丛中的均匀分布。这个方向的另一个新结果是建议的一部分：测地线流的遍历性可以用来确定特征函数的复节点超曲面的渐近性。在另一个方向上，特征函数的超范数与测地线流的焦点的存在以及焦点处的第一个返回的Mapon测地线方向有关(与C。Sogge和J.Toth)。渐近几何旨在应用类似的方法来研究复杂几何中的问题，特别是Yau，Tian和Donaldson的程序，研究正线丛上的所有Hermite度量用Bergman度量逼近。与J.Song的一个合作项目是利用环形簇上的Bergman核的渐近性来研究Phong-Sturm关于高度为N的Bergman度量的对称空间的几何如何逼近Hermitian度量的全无限维空间的Monge-Ampere测地几何的问题。在另一个方向上，与B.Shiffman正在进行的一个联合项目是研究高次变量的统计代数几何，因为次数趋于无穷大。这在弦理论中有应用(与M·道格拉斯共同工作)。这两个主题都涉及关于经典和量子力学共同测量和分析关系的思想和方法的应用。在每个区域都有一个接近于零的小“普朗克常数”。在本征函数的整体分析中，它是对本征值的1；在几何中，它是多项式的次数。当“普朗克常数”趋于零时，像本征函数或伯格曼核这样的分析对象(量子力学)倾向于几何对象(经典力学)，这更容易理解。近百年来，物理学家、工程师和数学家一直在研究经典力学和量子力学之间的关系，但这些关系是如此之难，以至于基本问题仍然存在。举一个古老的例子，Chladnyi在200年前首次证明，如果一个人把沙子放在振动的滚筒上，沙子就会移动到“节线”上。尽管有两百年的历史，但没有人知道普通鼓上的节线是如何蜿蜒的。上面的一个项目是，当台球以混乱的方式在鼓上移动时，确定“复杂的”节线。因此，如果一个人在复杂的方向上“加厚”结节，就可以理解它是如何在混乱的鼓头上蜿蜒前进的。

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.