GEOMETRIC DIFFERENTIAL EQUATIONS AND APPLICATIONS

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ABSTRACTGEOMETRIC DIFFERENTIAL EQUATIONS AND APPLICATIONSP.I.: Gang Tian (M.I.T.)This proposal concerns existence and regularity problems of the Einstein equation, the Yang-Mills equation on Riemannian manifolds and pseudo-holomorphic curves, as well as applications of these equations to geometry and topology. For the Einstein equation, we will focus mainly on (1) the existence of its solutions, particularly, K\"ahler-Einstein metrics in complex geometry and (2) behavior of its singular solutions. I will also study the associated K\"ahler-Ricci flow and its soliton-type solutions. The Yang-Mills equation is a nonlinear equation and may have singular solutions. We will study the basic problem how singular solutions behave along their singularity, e.q., the size of singularity. It has been found in my previous works that there is a connection between singularity formation of Yang-Mills fields and classical minimal submanifolds. I intend to explore this more and its related compactness problem for Yang-Mills fields, particularly, the interaction between self-dual solutions the Yang-Mills equation and calibrated geometry. I also intend to continue his study in symplectic geometry. The problems include symplectic isotopy problem in a rational surface and its applications toward classifying symplectic four dimensional spaces, computing well-defined symplectic invariants, such as the Gromov-Witten invariants, constructing new deformation invariants.Problems in this proposal arose naturally from our attempts to understanding nonlinear differential equations from geometry and physics. These equations include static Einstein equation, Yang-Mills fields as well as holomorphic maps. They played a fundamental role in our understanding of nature through mathematical means. They also have found many deep applications in geometry and topology, such as Seiberg-Witten theory, Mirror symmetry of Calabi-Yau spaces. The resolution of these problems will provide mathematical foundations for some physical theories and 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 differential equations which describe those phenomena. It is still challenging to have a complete mathematical understanding of these singular solutions. This proposal will address some of these basic problems. I will also try to find more solutions of these nonlinear equations -and apply them to studying basic problems in geometry and topology.