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

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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.