Heat Kernal Analysis and Analytic Number Theory on Symmetric Spaces, Calabi-Yau Varieties and Moduli Spaces

点击翻译按钮获取中文摘要

This award supports research in the general area of spectral theory and number theory, with particular interest in trace formulae, explicit formulae and prime number theorems, spectral theory on degenerating Riemannian manifolds, and various analytic and arithmetic aspects of Arakelov theory on Calabi-Yau varieties. Jorgenson is continuing his collaboration with Serge Lang. They are studying the heat kernel on symmetric spaces to provide further examples of their general theory. Specific examples such as Hilbert modular and Picard modular varieties will be studied deeply. Jorgenson is continuing resarch with Andrey Todorov in the study of the spectral theory and arithmetic of Calabi-Yau varieties. The long term goal of this study is the development of a theory of arithmetic of Calabi-Yau varieties which would be similar to the existing theory of arithmetic of elliptic curves. With Jozek Dodziuk, Jorgenson is studying the spectral theory of degenerating families of finite volume Riemannian manifolds which possess metrics that have constant negative sectional curvatures near the developing singularities. Jorgenson is continuing ongoing research with Jonathon Huntley in the study of Weyl's laws for the counting function of cuspidal eigenvalues on quotients of symmetric spaces of arbitrary rank, with particular emphasis on the symmetric spaces associated to GL(n,R). In collaboration with Jurg Kramer, Jorgenson is continuing the investigation of the Arakelov theory of Calabi-Yau varieties. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data tran smission and processing, and communication systems.