Geometry and Analysis of Locally Symmetric Spaces and Moduli Spaces of Riemann Surfaces

这个建议的目的是研究局部对称空间的分析和几何以及黎曼曲面的模空间，具有标准度量的黎曼曲面是局部对称空间的重要例子。虽然它们的模空间一般不是局部对称空间，但人们一直认为它们与局部对称空间有许多相似的性质，实际上，根据这一原理，模空间的许多拓扑性质已经被发现。这一建议旨在进一步研究它们之间的相似性，理解和促进局部对称空间和黎曼曲面模空间的共同性质，例如，局部对称空间的谱理论是自守形式的基础，著名的Langlands纲领，这个建议的一个项目是理解黎曼曲面模空间的谱理论和几何散射理论，完整的黎曼度量。另一个项目是理解高斯-博内公式和模空间的指数理论。算术群的约化理论对于理解局部对称空间和二次型是至关重要的，该提议还建议研究黎曼曲面模空间的类似约化理论。对称性是一个非常重要的概念，在科学和艺术中起着基础性的作用。它不仅有效而且美观。例如，物理学和自然界的许多基本定律都是从对称性原理推导出来的，美丽的造型和设计也遵循对称性原理，对称性的数学语言就是群论和相关的对称空间。也许最对称的空间是欧几里德空间。事实上，它是一类重要的空间称为对称空间，其中包括双曲空间和领域。商的对称空间被称为局部对称空间，他们是密切相关的另一类重要的空间在数学，模空间分类数学对象。

This proposal aims to study analysis and geometry of locally symmetric spaces and the moduli space of Riemann surfaces.Riemann surfaces with canonical metrics are important examples of locally symmetric spaces. Though their moduli spaces are not locally symmetric spaces in general, it has been recognized for a long time that they share many similar properties with locally symmetric spaces.Indeed, many topological properties of the moduli spaces have been found by following this principle. This proposal proposes to pursue further the analogues between them and understand and contribute more to common properties of locally symmetric spaces and the moduli spaces of Riemann surfaces.For example, the spectral theory of locally symmetric spaces is fundamental in automorphic forms and the celebrated Langlands program, and one project of this proposal is to understand the spectral theory and the geometric scattering theory of the moduli spaces of Riemann surfaces with respect to some canonical complete Riemannian metrics.Another project is to understand the Gauss-Bonnet formula and the index theory of the moduli spaces. The reduction theory for arithmetic groups is crucial for understanding locally symmetric spaces and quadratic forms, and this proposal also proposes to study analogous reduction theory for the moduli spaces of Riemann surfaces.Symmetry is a very important concept and has played a fundamental role in science and art. It is not only effective but also beautiful.For example, many basic laws in physics and nature are derived from the principle of symmetry, and beautiful forms and designs also follow the principle of symmetry.The mathematical language of symmetry is group theory and related symmetric spaces. Probably the most symmetric space is the Euclidean space.Indeed it is one of the important class of spaces called symmetric spaces, which include the hyperbolic spaces and spheres.Quotients of symmetric spaces are called locally symmetric spaces and they are closely related to another important class of spaces in mathematics, moduli spaces which classify mathematical objects.