Spectral Theory and Geometry of Locally Symmetric Spaces

谱论与局部对称空间几何

• 批准号: 0072299
• 负责人: Lizhen Ji
• 金额: $ 7.21万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072299&HistoricalAwards=false
• 关键词: Spectral; Theory; Geometry; Locally; Symmetric

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.

对称空间和局部对称空间是数学中的重要对象，并且来自于许多不同的学科，如李群理论，表示论，数论，微分几何，代数几何和动力学。 许多这样的自然空间是非紧的。例如，行列式1的正定矩阵空间是一个非紧对称空间，所有椭圆曲线的模空间是一个有限体积的非紧局部对称空间，这是志村曲线的一个例子，在最近的费马定理的解决方案中起着重要作用。 为了理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。 本计划中四个项目的共同主题之一是理解紧化的精细结构及其与空间谱理论的关系。例如，在局部对称空间中，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本征函数，特别是散射矩阵。对称空间的紧化在理解不变微分算子的联合本征函数和表示的矩阵系数在无穷远处的行为中起着重要的作用。 整体对称空间和局部对称空间的紧化以前主要是分别研究的，而这个建议的一个重要特点是用类似的方法研究这两种空间的紧化。数学家研究几何形状和它们的结构。 这样的形状集合由称为流形的对象组成。如果一个鼓被描绘成一种特殊类型的流形，那么由鼓产生的音调可以被认为是与流形相关的物质对象。 对于一个特别重要的收集鼓有两种音调：离散（或孤立）的和连续的家庭。PI打算研究这些鼓或歧管上的各种数学结构。