Mathematical Sciences: Analysis On Locally Symmetric Spaces

摘要 斯坦顿DMS-9624387 斯坦顿将继续他对光谱不变量的研究 与椭圆微分算子相关的各种设置， 齐性和局部对称空间。特别是，他应 检验全纯向量的全纯挠率公式 局部对称空间上的丛，在早期的工作中得到， 为了在其中隔离由下式定义的R类的贡献， 比斯穆特和吉莱-苏尔他将尝试扩展他早期的工作 在局部对称空间，其中他有关扭转特殊 值的几何zeta函数，一类紧凑，Kaehler， 齐性空间对于四元数-Kaehler对称 和局部对称空间，他将寻求一个新的谱不变量， 他将尝试建立一个相关的几何zeta函数 来检测这个不变量。在另一个研究方向上，他 他将继续研究 使用几何表示理论的真实的旗流形。 继60年代指数定理的成功之后， 提供了一个分析框架来计算拓扑不变量 流形，从20世纪70年代开始，谱不变量是 与椭圆微分算子相关联， 用公式表达更微妙的拓扑和几何不变量， 流形的语言分析，即微分方程， 在歧管上。这些新的不变量的重要性是 在他们的出现强调在新兴的理论， 理论物理学家的基本力虽然这些不变量 对于一般空间来说很难计算，因为这些空间 从群的作用，如局部对称空间， 谐波分析来自发达的理论， 群的表示提供了一个强大的工具来研究 这些不变量。几何zeta函数的引入 以一种尚未被理解的方式连接流形的拓扑结构， 通过基本群，通过以下值， zeta函数在不同的点。目标是理解 微分方程的拓扑性质，然后开始 一个与数论接触的正式层面。

Abstract Stanton DMS-9624387 Stanton will continue his investigation of spectral invariants associated to elliptic differential operators in various settings of homogeneous and locally symmetric spaces. In particular, he shall examine a formula for the holomorphic torsion of a holomorphic vector bundle over a locally symmetric space, obtained in earlier work, in order to isolate in it the contribution from the R-class defined by Bismut and Gillet-Soule. He will attempt to extend his earlier work on locally symmetric spaces, in which he related torsion to special values of geometric zeta functions, to a class of compact, Kaehler, homogeneous spaces. For the class of quaternionic-Kaehler symmetric and locally symmetric spaces he will seek a new spectral invariant, and he will attempt to develop an associated geometric zeta function to detect this invariant. In a different direction of research, he will continue his investigation of various topological properties of real flag manifolds using geometric representation theory. Following upon the success of the index theorem in the 60's to provide an analytic framework to compute topological invariants of manifolds, beginning in the 1970's, spectral invariants were associated to elliptic differential operators in an attempt to formulate more subtle topological and geometric invariants of manifolds in the language of analysis, i.e. differential equations, on the manifold. The importance of these newer invariants is underscored in their appearance in the emerging theory of the fundamental forces by theoretical physicists. While these invariants are difficult to compute for general spaces, for those spaces arising from the action of a group, such as locally symmetric spaces, the harmonic analysis coming from the well-developed theory of the representations of the group provides a powerful tool to investigate these invariants. The introduction of geometric zeta functions serves to connect in a not-yet-understood way the topology of the manifold, via the fundamental group, to these invariants through the values of the zeta function at distinguished points. The goal to understand topological properties through differential equations then begins on a formal level to make contact with number theory.