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

斯坦顿教授的研究项目将应用非对易调和分析和算子行列式来理解厄米局部对称流形上的全纯扭转。这项工作与H. Moscovici合作，将重点放在项目的当前阶段，即构建与一般秩空间相关的zeta函数，并以几何形式评估它们的特殊值。另一个项目，与A. Knapp共同开始，将研究全局对称流形上的某些微分算子。本课题将尝试给出这些算子的解空间的表示理论表征，从而解决Vogan的一个猜想。流形是对二维光滑曲面的高维的推广。例如，弯曲的时空连续体是一个四维流形。斯坦顿教授的研究旨在通过利用与某些偏微分方程的联系来理解这些几何物体。

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.