Geometry of the Moduli Spaces of Complex Manifolds

Calabi-Yau流形、Riemann曲面和一般类型流形的模空间、稳定丛和稳定映射在从几何、拓扑学、代数几何到数论的许多数学学科中都起着重要的作用，在弦论中也是重要的。镜像对称性激发了许多深刻的数学结果和理论。在这个项目中，PI将研究这些模空间的几何的各种局部和全局问题。这些问题包括正则度量的变形，如Kahler-Einstein度量和Bergman度量相对于各种度量。这项研究导致了Weil-Petersson型度量的几何量子化。PI还将研究从TeichMuller空间和模空间到Hodge结构及其商的变异分类空间的周期映射的全局性质。PI还将研究这些模空间上的标准Kahler度量和新Kahler度量的几何性质，以及非紧流形和它们上的丛上Kahler-Einstein和Hermitian-Einstein度量的普遍存在性。Kahler-Einstein流形上的TeichMuller和模空间对于数学和理论物理都是重要的。在弦论中，许多基本配分函数可以归结为模空间的几何或拓扑不变量，这也激发了许多美丽的猜想。对这些重要猜想的研究不仅促进了数学各分支的发展，而且有助于高能物理理论的验证。PI的研究在医学图像处理、3D摄像机等工业领域也具有潜在的应用前景。例如，通过显式构造周期图，可以显著提高压缩/解压缩画面的速度。

Moduli spaces of Calabi-Yau manifolds, Riemann surfaces and general type manifolds, stable bundles and stable maps have played fundamental roles in many subjects of mathematics from geometry, topology, algebraic geometry, to number theory and are also important in string theory. Mirror symmetry has motivated many deep mathematical results and theories. In this project the PI will study various local and global problems of the geometry of these moduli spaces. These problems include the deformation of canonical metrics such as the Kahler-Einstein metrics and Bergman metrics with respect to various gauges. This study lead to a geometric quantization of the Weil-Petersson type metrics. The PI will also study the global properties of the period map from the Teichmuller and moduli spaces to the classifying spaces of variation of Hodge structures and their quotients. The PI will also investigate the geometric properties on canonical and new Kahler metrics on these moduli spaces as well as the general existence of Kahler-Einstein and Hermitian-Einstein metrics on noncompact manifolds and bundles over them.Teichmuller and moduli spaces of Kahler-Einstein manifolds are important for both mathematics and theoretical physics. In string theory, many fundamental partition functions can be reduced to geometric or topological invariants of moduli spaces which also motivated many beautiful conjectures. The study of these important conjectures not only enhanced the development of various branches of mathematics but also help in verifying theory in high energy physics. The research of the PI also has potential applications in industry such as medical image processing and 3D cameras. For example, with the explicit construction of the period map the speed of compressing/decompressing pictures can be significantly increased.