Cohomology of Locally Symmetric Spaces

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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.