Geometry and Analysis on Calabi-Yau and Hermitian Manifolds

摘要奖：DMS 1308988，首席研究员：华伦天奴·托萨蒂PI建议使用非线性偏微分方程组研究关于复流形和辛流形的几个问题。第一个项目是关于了解Ricci-Flat Calabi-Yau流形在家族中退化的方式。在他以前工作的基础上，PI建议理解这些退化，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关于超kahler流形镜像对称的Strominger-Yau-Zaslow图的猜想。在第二个项目中，PI将使用Chern-Ricci流研究厄米特流形的几何，Chern-Ricci流是Kahler-Ricci流到所有复杂流形的扩展。这种流动与流形的复杂结构密切相关，并将用于拓宽我们对非Kahler紧致复杂曲面的理解。第三个项目是以Donaldson的程序为中心，将Kahler几何中Calabi猜想的Yau解推广到辛四维流形。这将为构造一类高度非线性偏微分方程解的闭辛四维流形提供一种新的、强有力的分析工具，并可用于解决辛拓扑中的基本公开问题，如：给定一个紧致的几乎复的四维流形，什么时候有相容的辛形？在这个领域，人们研究几何性质的问题(例如，称为流形的高维空间是如何弯曲的)，使用分析和微分方程的工具。本文的主要研究对象之一是Calabi-Yau流形。根据弦理论，我们的物理时空不是四维的，而是十维的。剩下的六个维度非常小，所以我们通常无法感知它们，但它们对于理解基本粒子是至关重要的。这六个维度一起形成了一个微小的几何空间，这是一个卡拉比-尤流形，它捕捉到了粒子物理学的基本特征。了解它的几何结构将使我们能够了解粒子是如何产生的，以及它们是如何相互作用的，这是当前数学物理中的主要问题之一。

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.