Geometric Analysis on Moduli Spaces of Riemann Surfaces and Locally Symmetric Spaces

紧致黎曼曲面或带穿孔的紧致黎曼曲面的模空间是数学中最重要的空间之一，在代数几何、复几何、拓扑学和数学物理等领域得到了广泛的研究。一个密切相关的空间是映射类群作用的黎曼曲面的Teichmueller空间，其商等于黎曼曲面的模空间。映射类群是几何群论中研究较多的一个基本群。Teichmueller空间以及黎曼曲面的模空间允许几种自然黎曼度量，如Weil-Petersson度量，Ricci度量和庞加莱型度量。本研究的主要目的是研究模空间的谱理论与这些黎曼度量，并了解其与模空间的几何和拓扑的关系。模空间与局部对称空间、映射类群与算术群之间的相似性的研究已经取得了丰硕的成果，本文的第二个目标是研究局部对称空间的一些相关问题。具体而言，本文包括以下4个项目：(1)模空间上不完全Weil-Petersson度量的谱理论。(2)模空间上完全黎曼度量的光谱理论：几何散射理论。(3)模空间的简单体积和简单棘。(4)对称空间的等变棘和局部对称空间的L^p谱理论。鼓对应于平面上的一个域，其音调对应于具有狄利克雷边界条件的域的拉普拉斯算子的特征值。一个可能很幼稚的问题是这些特征值如何与鼓的形状相关并反映鼓的形状，即域的几何形状。例如，一个非常大的鼓音调很低。马克·卡茨（Marc Kac）在1966年提出了一个著名的问题：“人能听到鼓的形状吗？”这个问题一直是光谱几何学科的推动力之一。定域是黎曼流形的特殊例子，数学家们一直在试图理解各种黎曼流形的几何和光谱理论，例如三维欧几里德空间内的封闭曲面。一类重要的空间来自具有相似性质的数学对象的集合，即所谓的模空间。本建议的主要目的是了解黎曼曲面的模空间的几何和谱。

The moduli space of compact Riemann surfaces or compact Riemann surfaces with punctures is one of the most important spaces in mathematics and has been extensively studied in algebraic geometry, complex geometry, topology and mathematics physics. A closely related space is the Teichmueller space of Riemann surfaces where the mapping class group acts, and the quotient is equal to the moduli space of Riemann surfaces.The mapping class group is a basic group intensively studied in geometric group theory. The Teichmueller space and hence the moduli space of Riemann surfaces admit several natural Riemannian metrics such as the Weil-Petersson metric, the Ricci metric and Poincare type metrics. The main goal of this proposal is to study the spectral theory of the moduli space with respect to these Riemannian metrics and to understand its relations with the geometry and topology of the moduli space. Pursuing similarities between the moduli space and locally symmetric spaces, and between mapping class groups and arithmetic groups has been fruitful, the second goal of this proposal is to study some related problems for locally symmetric spaces. Specifically, the proposal consists of the following 4 projects:(1) Spectral theory of the incomplete Weil-Petersson metric on the moduli space.(2) Spectral theory for complete Riemannian metrics on the moduli space: geometric scattering theory.(3) Simplicial volume and spines of the moduli space.(4) Equivariant spines of symmetric spaces and L^p-spectral theory of locally symmetric spaces.A drum corresponds to a domain in the plane, and its tones correspond to the eigenvalues of the Laplace operator of the domain with the Dirichlet boundary condition. A perhaps naive question is how these eigenvalues are related to and reflect the shape of the drum, i.e., the geometry of the domain. For example, a very large drum has a low pitch. A famous question raised by Marc Kac in 1966 is "Can one hear the shape of a drum?". This question has been one of the motivating forces for the subject of spectral geometry. Domains are special examples of Riemannian manifolds, and mathematicians have been trying to understand geometry and spectral theory of various Riemannian manifolds, for example, closed surfaces inside the three dimensional Euclidean space. An important class of spaces comes from collections of mathematical objects sharing similar properties, the so-called moduli spaces. The main purpose of this proposal is to understand the geometry and spectral of moduli spaces of Riemann surfaces.