Geometry and Analysis of Differentiable Manifolds

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The problems investigated in this project concern various nonlinear differential equations from geometry and physics. These equations are of the evolution type and involve the curvature which measures how a space is curved. They play a fundamental role in understanding the natural world through mathematical means and are closely related to the study of the field theories in physics. They have also found many deep applications in geometry and topology. A famous example is Perelman's solution of the Poincare conjecture by using Ricci flow. The resolution of the problems in this project will provide mathematical foundations for some physical theories and have further profound applications to long-standing mathematical problems such as the classification of algebraic spaces. The most common phenomena of these equations are their singular behaviors due to the nonlinearity of the equations. Such behaviors are reflected in the possible break-downs in the evolution process and described in terms of singular solutions to these equations which describe the evolution process. It is still challenging to have a complete mathematical understanding of these singular solutions. This project will address some basic problems on these singular solutions and explore their applications to geometry and topology. The PI will give lectures and teach graduate courses on topics directly related to this project. He will also run a geometry working seminar with a goal of helping students to gain research experiences and broaden their knowledge in mathematics.This project concerns curvature flows and equations in Riemannian geometry. 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 formation and geometry of the underlying spaces; (3) The long-time behavior of the solutions. For the Hermitian curvature flow, the PI will develop new analytic tools to study how it forms singularity. One of most prominent Hermitian curvature flow is the pluriclosed flow which is connected to the renormalization group flow of the nonlinear sigma model with B-field. The PI will further explore this connection and gives new mathematical insights for the duality in the string theory on one hand, new understanding of finite-time singularity on the other hand. For the symplectic curvature flow, the PI intends to study how to characterize the maximal existence of the flow by cohomological condition and how it develops finite-time singularity in dimension 4. The PI also intends to extend the compactness theory for Einstein metrics to a more general class of Kahler metrics and 4-dimensional anti-self-dual metrics. He also continues his study on fundamental problems in symplectic geometry which involve certain gauge equation. The problems include constructing new deformation invariants for symplectic manifolds which admit a Hamiltonian S1-action and providing a mathematical theory for the gauged linear sigma model. These problems are important in symplectic geometry and are inspired by the topological field theories in physics.