On the Geometry of Moduli Space and Kahler-Einstein Geometry

研究者将研究微分几何中的基本问题。虽然该研究项目涉及纯数学中的微分几何，但其方法和结果对我们对宇宙的理解产生了重大影响。特别是，这个项目研究弦理论的数学方面的深层问题，弦理论是当前理论物理学中一个非常感兴趣的话题。研究者参与了K-12教育活动，也深入参与了本科和研究生课程的创新。PI将研究卡勒几何和光谱理论中的几个重要问题。该项目将继续对Calabi-Yau模上的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Bergman核的Agmon型估计，并将继续研究完全非紧流形上微分形式的本质谱。该研究将建立在PI及其合作者在复杂几何和光谱理论方面的最新成果的基础上，包括证明Kahler流形模空间上的chen - weil形式的合理性；一类有规流形上极值度量的构造非紧致完全Kahler流形上Bergman核的非对角展开和Bergman核的Poincare级数的研究Antunes-Freitas猜想的证明；研究狄利克雷特征值与诺伊曼特征值之间的关系。这个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.