Geometry and Analysis on Calabi-Yau and Hermitian Manifolds

Calabi-Yau 和 Hermitian 流形的几何与分析

• 批准号: 1308988
• 负责人: Valentino Tosatti
• 金额: $ 19.14万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308988&HistoricalAwards=false
• 关键词: Geometry; Analysis; Calabi; Yau; Hermitian

AbstractAward: DMS 1308988, Principal Investigator: Valentino TosattiThe PI proposes to investigate several problems about the geometry of complex and symplectic manifolds using nonlinear partial differential equations. The first project is about understanding the ways in which Ricci-flat Calabi-Yau manifolds can degenerate in families. Building on his previous work, the PI proposes to understand these degenerations, to explore the structure of the possible limit spaces, and to apply these results to attack a conjecture of Kontsevich-Soibelman, Gross-Wilson and Todorov related to the Strominger-Yau-Zaslow picture of mirror symmetry for hyperkahler manifolds. In the second project the PI will study the geometry of Hermitian manifolds using the Chern-Ricci flow, an extension of the Kahler-Ricci flow to all complex manifolds. This flow is intimately related to the complex structure of the manifold and will be used to widen our understanding of non-Kahler compact complex surfaces. The third project is centered on Donaldson's program to extend Yau's solution of the Calabi Conjecture in Kahler geometry to symplectic four-manifolds. This would provide a new and powerful analytic tool to construct symplectic forms on closed symplectic four-manifolds as solutions of a highly nonlinear PDE, and would allow to solve basic open questions in symplectic topology, such as: given a compact almost-complex four-manifold, when are there compatible symplectic forms?The proposed research is in the field of Geometric Analysis. In this area one studies problems of geometric nature (for example how a high-dimensional space, called a manifold, is curved), using the tools of analysis and differential equations. One of the main objects of study in the proposed research are Calabi-Yau manifolds. According to string theorists, our physical space-time is not four-dimensional but rather ten-dimensional. The remaining six dimensions are extremely small, so that we don't normally perceive them, but are crucial for understanding elementary particles. These six dimensions together form a tiny geometric space, which is a Calabi-Yau manifold, and which captures essential features of particle physics. Understanding its geometry would allow us to understand how particles are created and how they interact, and is one of the main current problems in mathematical physics.

摘要奖项：DMS 1308988，首席研究员：Valentino Tosatti PI 提议使用非线性偏微分方程研究有关复辛流形几何的几个问题。第一个项目是关于了解 Ricci-flat Calabi-Yau 流形在族中退化的方式。在他之前的工作基础上，PI 提议理解这些退化，探索可能的极限空间的结构，并应用这些结果来攻击 Kontsevich-Soibelman、Gross-Wilson 和 Todorov 与 Strominger-Yau 相关的猜想。超卡勒流形镜像对称的 Zaslow 图片。在第二个项目中，PI 将使用 Chern-Ricci flow（将 Kahler-Ricci flow 扩展到所有复杂流形）来研究 Hermitian 流形的几何形状。这种流动与流形的复杂结构密切相关，将用于扩大我们对非卡勒紧凑复杂表面的理解。第三个项目以唐纳森的计划为中心，将丘的卡勒几何中卡拉比猜想的解扩展到辛四流形。这将提供一种新的强大的分析工具来在闭辛四流形上构建辛形式作为高度非线性偏微分方程的解，并且将允许解决辛拓扑中的基本开放问题，例如：给定一个紧凑的几乎复杂的四流形流形，何时存在兼容的辛形式？拟议的研究属于几何分析领域。在这一领域，人们使用分析和微分方程工具来研究几何性质的问题（例如，称为流形的高维空间如何弯曲）。本研究的主要研究对象之一是 Calabi-Yau 流形。根据弦理论家的说法，我们的物理时空不是四维的，而是十维的。其余的六个维度非常小，因此我们通常无法感知它们，但对于理解基本粒子至关重要。这六个维度共同形成一个微小的几何空间，即卡拉比-丘流形，它捕捉了粒子物理学的基本特征。了解它的几何形状将使我们能够了解粒子是如何产生的以及它们如何相互作用，这是数学物理学中当前的主要问题之一。