Spectral Theory and Geometry of Locally Symmetric Spaces

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DMS-0072299Lizhen Ji Symmetric and locally symmetric spaces are important objects in mathematics and arise from many different subjects such as Lie group theory, representation theory, number theory, differential geometry, algebraic geometry, and dynamics. Many natural such spaces are noncompact. For example, the space of positive definite matrices of determinant one is a noncompact symmetric space, and the moduli space of all elliptic curves is a noncompact locally symmetric space of finite volume which is one example of Shimura curves and plays an important role in the recent solution of the last Fermat's theorem. To understand the geometry and analysis of such noncompact spaces, an important problem is to study their compactifications. One of the common themes of the four projects in this proposal is to understand refined structures of the compactifications and their relations to the spectral theory of the spaces. For example, for locally symmetric spaces, geodesics which are eventually distance minimizing can be used to study the compactifications and also to understand the generalized eigenfunctions of the continuous spectrum, specifically, the scattering matrices.Compactifications of symmetric spaces play an important role in understanding behaviors at infinity of the joint eigenfunctions of the invariant differential operators and the matrix coefficients of representations. The compactifications of globally and locally symmetric spaces have mainly be studied separately before, and an important feature of this proposal is to study compactifications of both types of spaces using a similar approach.Mathematicians study geometric shapes and their structures. Onesuch collection of shapes consists of objects called manifolds. If a drum is pictured as a particular type of manifold then thetones produced by the drum can be thought of as mathematicalobjects associated to the manifold. For a particularly importantcollection of drums there are two kinds of tones: the discrete (orisolated) ones and the continuous families. The PI intends to investigate a variety of mathematical structures on these drums or manifolds.