Cohomology of Locally Symmetric Spaces

摘要：本课题研究非紧局部对称空间的上同调问题。在之前的工作中，研究者在局部对称空间的紧化上开发了纯度的微局部类似物，并用它来证明Rapoport猜想。在目前的项目中，研究者将应用这一纯度理论来研究局部对称空间的上同调与其边界层的上同调之间的关系。一个工具将是由研究者开发的权重谱序列，它是基于朗兰兹划分的。目的是从更几何的角度理解用爱森斯坦级数构造上同调及其与l函数的关系。局部对称空间自然出现在数学和理论物理的许多领域，特别是几何和数论。在他们最基本的形式，几何和数论在高中被考虑：平面图形的研究和整数的研究。这是数学中最古老的两个领域。从现代的角度来看，作为朗兰兹计划的一部分，几何和数论交织在一起，在过去的四分之一世纪里，一些最令人兴奋的研究都是在朗兰兹计划中进行的。几何学和数论的应用比比皆是，特别是在编码理论、计算和晶体、材料和生物结构的对称性研究方面。

Abstract NSF Proposal: DMS 9870162 Principal Investigator: Leslie D. Saper This project deals with the cohomology of noncompact locally symmetric spaces. In previous work, the investigator developed a micro-local analogue of purity on compactifications of locally symmetric spaces and used this to prove Rapoport's conjecture. In the current project, the investigator will apply this purity theory to study the relation between the cohomology of a locally symmetric space and the cohomology of its boundary strata. One tool will be a weight spectral sequence developed by the investigator which is based on the Langlands's partition. The objective is to understand from a more geometric viewpoint the construction of cohomology using Eisenstein series and the relation with L-functions. Locally symmetric spaces occur naturally in many areas of mathematics and theoretical physics, in particular, geometry and number theory. In their most basic forms, geometry and number theory are considered in high school: the study of plane figures and the study of the whole numbers. These are two of the oldest fields in mathematics. In the modern perspective, geometry and number theory become intertwined as part of the Langlands's program, the scene of some of the most exciting current research of the past quarter-century. Applications of geometry and number theory abound, in particular in coding theory, computing, and the study of symmetry of crystals, materials, and biological structures.