Mathematical Sciences: Geometric Analysis and Spectral Invariants on Locally Symmetric Manifolds

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Professor Stanton's research project will apply noncommutative harmonic analysis and operator determinants to gain an understanding of holomorphic torsion on Hermitian locally symmetric manifolds. This effort, in collaboration with H. Moscovici, will focus on the current phase of the project which is to construct zeta functions associated to general rank spaces and to evaluate their special values in geometric terms. Another project, begun jointly with A. Knapp, will investigate certain differential operators on globally symmetric manifolds. This project will attempt to give a representation theoretic characterization of the solution space of these operators thereby resolving a conjecture of Vogan. A manifold is a generalization to higher dimensions of a smooth two dimensional surface. For example, the curved space-time continuum is a four dimensional manifold. Professor Stanton's research is aimed at understanding these geometric objects by exploiting connections with certain partial differential equations.