Geometric Flows and Almost Complex Geometry

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

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.