Geometric Analysis on Complex Manifolds

复杂流形的几何分析

• 批准号: 1610278
• 负责人: Valentino Tosatti
• 金额: $ 22.5万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1610278&HistoricalAwards=false
• 关键词: Geometric; Analysis; Complex; Manifolds

The principal investigator's research concerns the study of complex manifolds, which are higher-dimensional curved spaces that are defined using the complex numbers. The simplest examples of such spaces are called Riemann surfaces, which are complex manifolds with one complex dimension (and therefore two real dimensions), and include the familiar surfaces of the sphere and of a donut. Higher-dimensional complex manifolds include for example Calabi-Yau manifolds, which are a fundamental tool in string theory. Complex manifolds are ubiquitous objects in mathematics, and have wide-ranging applications in physics and engineering. The proposed research projects will expand our knowledge of the geometry of higher-dimensional complex manifolds using analytic techniques, and in particular partial differential equations (PDEs). These projects lie at the intersection of several mathematical disciplines, such as differential, algebraic and symplectic geometry, complex analysis and PDEs, and techniques from all these fields are necessary to attack them. Progress on these questions will not only shed some light on some basic problems in mathematics, but will also have applications in physics and other sciences.The principal investigator proposes to use techniques from geometric analysis and nonlinear partial differential equations to investigate problems about the geometry of complex manifolds. The first project is about applications of analysis to the construction of currents on complex manifolds, which are used to study the geometry of (1,1) cohomology classes on compact Kahler manifolds. In the second project the principal investigator will develop new analytic techniques to construct special metrics on non-Kahler complex manifolds, by solving Monge-Ampere equations for (n-1,n-1) forms, building upon earlier work of the principal investigator with Szekelyhidi and Weinkove which culminated in the solution of Gauduchon's conjecture. The third project is about understanding collapsed limits of Ricci-flat Calabi-Yau manifolds. This is closely related to the theory of mirror symmetry, which was inspired by physical considerations. The fourth project is centered on Donaldson's program to extend Yau's solution of the Calabi Conjecture in Kahler geometry to symplectic four manifolds, and to its applications to symplectic topology.

首席研究员的研究涉及复流形的研究，复流形是使用复数定义的高维弯曲空间。这种空间的最简单的例子叫做黎曼曲面，它是具有一个复维（因此是两个真实的维）的复流形，包括我们熟悉的球面和圆环面。更高维的复流形包括例如卡拉比-丘流形，它是弦理论中的基本工具。复流形是数学中普遍存在的对象，在物理和工程中有着广泛的应用。拟议的研究项目将扩大我们的知识的几何高维复杂的流形使用分析技术，特别是偏微分方程（PDE）。这些项目位于几个数学学科的交叉点，如微分，代数和辛几何，复杂的分析和偏微分方程，以及来自所有这些领域的技术是必要的攻击他们。这些问题的研究进展不仅有助于揭示数学中的一些基本问题，而且还将在物理学和其他科学中得到应用。主要研究者建议使用几何分析和非线性偏微分方程的技术来研究复流形的几何问题。第一个项目是关于应用分析来构造复流形上的流，这是用来研究紧致Kahler流形上的（1，1）上同调类的几何。在第二个项目中，首席研究员将开发新的分析技术，通过求解Monge-Ampere方程的（n-1，n-1）形式，在首席研究员与Szekelyhidi和Weinkove的早期工作的基础上，在非Kahler复流形上构建特殊度量，最终解决了Gauduchon猜想。第三个项目是关于理解Ricci平坦Calabi-Yau流形的坍缩极限。这与镜像对称理论密切相关，镜像对称理论的灵感来自于物理考虑。第四个项目是集中在唐纳森的计划，以延长丘的解决方案的卡拉比猜想在Kahler几何辛四流形，及其应用辛拓扑。