Geometric Flows and Almost Complex Geometry

几何流和几乎复杂的几何

• 批准号: 1611797
• 负责人: Weiyong He
• 金额: $ 14.55万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1611797&HistoricalAwards=false
• 关键词: Geometric; Flows; Almost; Complex; Geometry

The so-called almost complex structure is a fundamental notion in the study of even dimensional spaces in mathematics. The understanding of almost complex structure in dimension four is of major importance in the mathematical branch of Differential Geometry. This has also great impact on other subjects including Physics. The projects designed aim to use powerful tools from geometric analysis to study almost complex structure, which will lead to applications in string theory, algebraic geometry and complex geometry. The proposed research will have immediate beneficial effects to students in the principal investigator's home university. The geometry of 4-dimensional smooth and symplectic manifolds has witnessed very exciting achievements in the last several decades. Despite the great success, the full understanding of smooth topology of four manifolds and symplectic manifolds remains largely open. In the last several decades, profound progress has also been made in the Ricci flow, for example, the solution of Poincare conjecture and geometrization conjecture in three manifolds. In this project, the PI will focus on compact almost complex manifolds by using a new kind of geometric flows, which evolve an almost complex structure to a symplectic structure. One motivation is to understand precisely when a compact four manifold with an almost complex structure supports a symplectic structure. Inspired by the Hamilton-Perelman's theory in the Ricci flow, the PI intends to study these new geometric flows in a systematical way, such as the longtime behavior and formation of singularities.

所谓几乎复数结构是数学中研究偶数维空间的一个基本概念。对四维几乎复杂结构的理解在微分几何的数学分支中具有重要意义。这对包括物理在内的其他学科也产生了很大的影响。设计的项目旨在使用几何分析的强大工具来研究几乎复杂的结构，这将导致在弦论、代数几何和复杂几何中的应用。这项拟议的研究将立即对首席调查员所在大学的学生产生有益影响。在过去的几十年里，4维光滑辛流形的几何学取得了令人振奋的成就。尽管已经取得了很大的成功，但对四种流形和辛流形的光滑拓扑的充分理解在很大程度上是开放的。在过去的几十年里，Ricci流也取得了深刻的进展，例如在三个流形上求解Poincare猜想和几何化猜想。在这个项目中，PI将通过使用一种新的几何流来关注紧致的几乎复杂的流形，它将几乎复杂的结构演化为辛结构。一个动机是精确地理解具有几乎复杂结构的紧凑的四个流形何时支撑辛结构。受Ricci流中的Hamilton-Perelman理论的启发，PI打算系统地研究这些新的几何流，如奇点的长期行为和形成。