On the Geometry of Moduli Space and Kahler-Einstein Geometry

University Of California-Irvine, Irvine CA

Investigators

Abstract

The investigator will work on fundamental problems in differential geometry. Although this research project concerns differential geometry, a subject in pure mathematics, the methods and results significantly influence our understanding of the universe. In particular, this project investigates deep problems in the mathematical aspects of string theory, a topic of great current interest in theoretical physics. The investigator is involved in K-12 educational activities and also deeply involved in innovations in both undergraduate and graduate curricula. The PI will work on several important problems in Kahler geometry and spectral theory. This project will continue a long-term investigation of the Weil-Peterson geometry on Calabi-Yau moduli, incorporating recent results in Kahler-Einstein geometry. The PI will study Agmon type estimates of the Bergman kernel and will continue to study the essential spectrum of differential forms on complete noncompact manifolds. The research will build on recent results of the PI and collaborators in complex geometry and spectral theory, including proof of the rationality of the Chern-Weil forms on moduli space of Kahler manifolds; construction of extremal metrics on certain ruled manifolds; study of the off-diagonal expansion of the Bergman kernels and the Poincare series of Bergman kernels on noncompact complete Kahler manifolds; proof of the Antunes-Freitas conjecture; and study of the relation between Dirichlet and Neumann eigenvalues. The long-term goal of this project is improved understanding of the geometry of moduli space; short term goals include study of the Bergman kernel expansion and the spectrum of differential forms on a complete non-compact manifold.