Bergman Kernel Estimates and Spectrum of Complete Riemannian Manifold

University Of California-Irvine, Irvine CA

Investigators

Abstract

In this project the Principle Investigator (PI) will work on some fundamental problems in a field of mathematics called differential geometry. Although differential geometry is branch of pure mathematics, the methods it employs and its results play a fundamental role in our understanding of the Universe. For example, String Theory, which is potentially the "theory of everything" in physics, uses a lot of tools from differential geometry. The PI plans to do work in the part of differential geometry which is closely related to string theory. The PI will also be involved in many K-12 educational activities and will be deeply involved in the creation and innovation of both undergraduate and graduate courses at the University of California at Irvine. The award will support graduate students working on their dissertations under the direction of the PI. More specifically, the PI plans to work on several important problems in Kahler geometry and spectral theory. In Kahler geometry he will continue his long term project on the analysis of the Bergman kernel and its relation to analysis on manifolds with bundle valued sections. In spectral theory, he will continue to study the essential spectrum of the Laplacian on differential forms on complete non-compact manifolds, and study inverse spectral problems. These two parts of work will be combined when the PI studies the geometry of Calabi-Yau moduli. In most cases, Calabi-Yau moduli are not complete manifolds, and therefore the PI needs to generalize the spectrum results onto incomplete manifolds before doing more non-linear analysis on them. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.