Geometric Flows and Almost Complex Geometry

所谓几乎复结构是数学中研究偶维空间的一个基本概念。对四维几乎复结构的理解在微分几何的数学分支中具有重要意义。这对包括物理学在内的其他学科也有很大的影响。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，这将导致弦理论，代数几何和复几何的应用。拟议的研究将对主要研究者所在大学的学生产生直接的有益影响。在过去的几十年里，四维光滑流形和辛流形的几何学取得了令人振奋的成就。尽管取得了巨大的成功，充分了解光滑拓扑的四流形和辛流形仍然在很大程度上开放。近几十年来，Ricci流也取得了很大的进展，如Poincare猜想和三维流形上的几何化猜想的求解。在这个项目中，PI将通过使用一种新的几何流来关注紧致几乎复流形，该几何流将几乎复结构演化为辛结构。一个动机是精确地理解具有几乎复结构的紧致四流形何时支持辛结构。受Ricci流中Hamilton-Perelman理论的启发，PI打算以系统的方式研究这些新的几何流，例如奇点的长期行为和形成。

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.