This project concerns existence and regularity problems of curvature equations in Riemannian geometry. These equations include the Ricci flow, the Einstein equation, the Yang-Mills equation. For Ricci flow, the PI will focus on (1) finite-time singularity formation for its solutions in Kahler geometry; (2) The interaction between the singularity and geometry of the underlying spaces; (3) long-time behavior of the solutions. The PI will study the pluri-closed flow, a new curvature flow which arises from complex geometry. He will develop its analytic theory and singularity formation. It has been found that there is a deep connection between this flow and the renormalization group flow of the nonlinear sigma model with B-field. The PI will explore this connection further and give new mathematical insights for the duality in the string theory. For the Einstein equation, the PI will focus mainly on the existence of its solutions and behavior of its singular solutions. The case for dimension 4 is particularly interesting. Self-dual metrics in dimension 4 will be also studied. They can be used to study the geometry and topology of underlying spaces. The PI will also study how the symplectic flow develops finite-time singularity in dimension 4 as well as other fundamental problems in symplectic geometry. The problems include isotopy of symplectic surfaces and its applications to classifying symplectic 4-manifolds. The PI will also continue his construction of new invariants for symplectic manifolds which admit a Hamiltonian S^1-action. These invariants are constructed by studying the Yang-Mills equation coupled with Cauchy-Riemann equation. These invariants and their extensions are inspired by the topological field theories in mathematical physics and provide new mathematical foundations for physical theories.
Problems in this project arose naturally from the attempts to understanding nonlinear differential equations from geometry and physics. These equations involve curvature and include Ricci flow, static Einstein equation. They played a fundamental role in understanding of nature through mathematical means. They also have found many deep applications in geometry and topology, such as the topology of low dimensional spaces. The resolution of these problems will provide mathematical foundations for some physical theories and will have profound applications to long-standing mathematical problems. Most natural phenomena are nonlinear and possess singular behaviors. These are reflected in possible singular solutions to the curvature equations which describe those phenomena. It is still challenging to have a complete mathematical understanding of these singular solutions. This project will address some of these basic problems.
