Mathematical Sciences: Analysis On Locally Symmetric Spaces

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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.