Canonical metrics on Kahler and Riemannian manifolds and their moduli

构造具有各种几何结构的适当空间是代数几何和微分几何中的一个基本问题。这些空间被称为模空间，在从分析、几何、拓扑学到数论的数学学科中扮演着重要的角色。一个多世纪前，Riemann、Poincare和Hilbert率先研究了表面上的“正则”公制结构。通过与爱因斯坦方程的联系，正在研究的项目将有助于理解天文学和宇宙学。本项目中分析爱因斯坦类型方程的方法也将在工程和经济学中得到应用。首席调查员组织和参与综合研究/教育方案，旨在吸引来自代表性不足群体的学生学习更高级的数学科目，如几何分析，从而帮助提高国家的整体教育水平。PI将研究以下项目，所有这些项目都来自于对模空间的代数/分析方面的研究。首先，他将通过代数和分析方法研究极化Kahler流形的模空间的几何(例如，模空间上的线丛和高度的正性)。其次，他将以PI及其合作者的工作为基础，研究Hilbert和Mumford发明的几何不变量理论(GIT)在代数模构造方面的有效范围。特别是，PI调查了更多的例子，以便开发一个比经典GIT更灵活的通用框架。第三，他将研究低维流形上Kahler-Ricci流的极限行为，特别是复杂结构跳跃的情况。第四，他将研究黎曼流形在无限维单位球面上的正则嵌入，并将嵌入的外在几何与基础黎曼流形上的正则度量联系起来。这是试图统一保角几何和卡勒几何之间相似性的第一步。上述提出的项目将应用来自代数/微分几何的分析和代数工具，以及主要研究人员和他的合作者多年来提出的新想法。

It is a fundamental problem to construct appropriate spaces of various geometric structures in both algebraic and differential geometry. These spaces, referred to as moduli spaces, have played a fundamental role in mathematical subjects ranging from analysis, geometry, and topology to number theory. More than a century ago, Riemann, Poincare, and Hilbert pioneered the study of "canonical" metric structures on a surface. Through the connection to the Einstein equations, the projects under investigation will help understand astronomy and Cosmology. The methods of analyzing Einstein type equations in this project will also lead to applications in Engineering and economics. The principal investigator organizes and participates in the integrated research/education programs that aim to attract students from under-represented groups to the study of more advanced mathematics subjects such as geometric analysis, thus helping to improve the overall education level of the nation. The PI will investigate the following projects, all of which emerge from the study of the algebraic/analytic aspects of the moduli spaces. First, he will study the geometry of the moduli space of polarized Kahler manifolds via algebraic and analytic means (e.g., the positivity of the line bundles and heights over the moduli space). Second, he will study the effective scope of geometric invariant theory (GIT) invented by Hilbert and Mumford on the construction of algebraic moduli based on the work of the PI and his collaborators. In particular, the PI investigates more examples in order to develop a general framework that is more flexible than the classical GIT. Third, he will study the limiting behavior of Kahler-Ricci flow on low-dimensional manifolds, in particular the situation when the complex structure jumps. Fourth, he will study the canonical embedding of Riemannian manifolds into the infinite-dimensional unit sphere and relate the extrinsic geometry of the embedding to the canonical metrics on the underlying Riemannian manifold. This is intended as the first step in attempting to unify the similarities between conformal and Kahler geometry. The projects proposed above will apply both analytic and algebraic tools coming from algebraic/differential geometry as well as the new ideas introduced by the principal investigator and his collaborators over the years.