Geometry and Analysis of Manifolds

Princeton University, Princeton NJ

Investigators

Abstract

Abstract for DMS-0804095 This proposal concerns geometric and analytic problems which arise from geometry and physics. The PI will continue his study on existence problem and singularity formation of the Einstein equation and the Yang-Mills equation. For the Einstein equation, the PI will focus mainly in dimension 4 or in the Kahler case. He will also study related Ricci flow and its singularity development and its interaction with classifying projective manifolds in algebraic geometry. Corresponding problems for self-dual metrics in dimension 4 will be also studied. For the Yang-Mills equation, the PI will focus on how to compactify spaces of its self-dual solutions and their applications to constructing new invariants. The PI found before that the Yang-Mills fields forms singularity along classical minimal surfaces like soap bubbles or subvarieties. He likes to explore this further and particularly, the interaction between self-dual Yang-Mills fields and calibrated geometry. The PI also intends to continue his study on problems in symplectic geometry, including deforming symplectic surfaces in 4-manifolds and constructing new deformation invariants. The Einstein and the Yang-Mills equations have played a fundamental role in our study of physics and geometry and topology in last few decades. Perelman's solution for the Poincare conjecture is an excellent example. Important and central problems include studying when one can solve those equations, what properties of those solutions found, how they develop singular behaviors. It is also important to understand the connection between these solutions and other branches of mathematics, such as, algebraic geometry and differential topology. The resolution of these problems will provide new profound understanding geometry of underlying spaces. The problems involved in this research project were also inspired by the study of the string theory in physics. Through this research project, the PI also intends to develop new tools for studying curved spaces, symplectic geometry and provide new mathematical foundation for some physical theories.