On the Geometry of Moduli Space and Kahler-Einstein Geometry

研究员将研究微分几何中的基本问题。虽然这个研究项目涉及微分几何，一个纯数学的主题，方法和结果显着影响我们对宇宙的理解。特别是，这个项目调查弦理论的数学方面的深层次问题，这是理论物理学当前非常感兴趣的一个主题。 调查员参与K-12教育活动，并深入参与本科和研究生课程的创新。PI将致力于Kahler几何和谱理论中的几个重要问题。该项目将继续对卡-丘模量的Weil-Peterson几何进行长期研究，并结合Kahler-Einstein几何的最新结果。PI将研究伯格曼核的Agmon型估计，并将继续研究完备非紧流形上微分形式的本质谱。该研究将建立在PI和合作者在复几何和谱理论方面的最新成果之上，包括证明Kahler流形模空间上的Chern-Weil形式的合理性;构造某些规则流形上的极值度量;研究非紧完备Kahler流形上Bergman核和Bergman核的Poincare级数的非对角展开;证明Antunes-Freitas猜想;研究了Dirichlet特征值与Neumann特征值之间的关系。该项目的长期目标是提高对模空间几何的理解;短期目标包括研究Bergman核展开和完全非紧流形上的微分形式谱。

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.