Spectral Theory and Geometry of Locally Symmetric Spaces

DMS-0072299李珍姬对称和局部对称空间是数学中的重要对象，产生于李群论、表示论、数论、微分几何、代数几何和动力学等多门学科。许多自然的这样的空间是不紧凑的。例如，行列式一的正定矩阵空间是非紧对称空间，所有椭圆曲线的模空间是有限体积的非紧局部对称空间，它是Shimura曲线的一个例子，在最近最后的Fermat定理的求解中起着重要的作用。要了解这种非紧空间的几何和分析，一个重要的问题是研究它们的紧化。本提案中四个项目的共同主题之一是理解压缩的精细结构及其与空间谱理论的关系。例如，对于局部对称空间，最终是距离最小化的测地线可以用来研究紧化，也可以用来理解连续谱的广义特征函数，特别是散射矩阵；对称空间的紧化对于理解不变微分算子的联合特征函数和表示的矩阵系数在无穷远处的行为有重要作用。以往对全局对称空间和局部对称空间的紧化研究主要是分开进行的，本文的一个重要特点是用相似的方法研究这两类空间的紧化。数学家研究几何形状及其结构。这样的形状集合由称为流形的对象组成。如果鼓被描绘成一种特定类型的歧管，那么由鼓产生的音调可以被认为是与歧管相关的数学对象。对于一个特别重要的鼓集合，有两种音调：离散的(或孤立的)音调和连续的家族。PI打算调查这些鼓或歧管上的各种数学结构。

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.