Geometry of the Moduli Spaces of Complex Manifolds

卡拉比-丘流形、黎曼曲面和一般类型流形、稳定丛和稳定映射的模空间在从几何、拓扑、代数几何到数论的许多数学学科中发挥了基础作用，并且在弦论中也很重要。镜像对称激发了许多深刻的数学结果和理论。在这个项目中，PI 将研究这些模空间几何的各种局部和全局问题。这些问题包括规范度量的变形，例如卡勒-爱因斯坦度量和伯格曼度量相对于各种规范的变形。这项研究导致了 Weil-Petersson 型度量的几何量化。 PI 还将研究从 Teichmuller 和模空间到 Hodge 结构及其商的变异分类空间的周期图的全局属性。 PI 还将研究这些模空间上的规范和新卡勒度量的几何性质，以及非紧流形及其束上的卡勒-爱因斯坦和埃尔米特-爱因斯坦度量的一般存在性。Teichmuller 和卡勒-爱因斯坦流形的模空间对于数学和理论物理都很重要。在弦理论中，许多基本的配分函数可以简化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。这些重要猜想的研究不仅促进了数学各个分支的发展，而且有助于验证高能物理理论。 PI的研究在医学图像处理和3D相机等工业领域也有潜在的应用。例如，通过周期图的显式构造，可以显着提高压缩/解压缩图片的速度。

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.