GEOMETRIC DIFFERENTIAL EQUATIONS AND APPLICATIONS

抽象几何微分方程及其应用田刚（M.I.T.）这个建议涉及的存在性和正则性问题的爱因斯坦方程，杨米尔斯方程黎曼流形和伪全纯曲线，以及应用这些方程的几何和拓扑。对于Einstein方程，我们主要讨论（1）它的解的存在性，特别是复几何中的K\“ahler-Einstein度量和（2）它的奇异解的行为。我们还将研究相应的Kahler-Ricci流及其孤子型解。Yang-Mills方程是一个非线性方程，可能有奇异解。我们将研究奇异解沿着其奇异性如何表现的基本问题，例如，奇点的大小。在我以前的工作中发现，杨-米尔斯场的奇点形成与经典极小子流形之间存在着联系。我打算探讨这个更多的和它的相关的紧性问题的杨米尔斯场，特别是，自对偶解之间的相互作用的杨米尔斯方程和校准几何。我也打算继续他的研究辛几何。这些问题包括有理曲面上的辛合痕问题及其在分类辛四维空间、计算定义良好的辛不变量（如Gromov-Witten不变量）、构造新的变形不变量等方面的应用，这些问题的提出是从几何和物理角度理解非线性微分方程的一个自然的尝试。这些方程包括静态爱因斯坦方程、杨-米尔斯场以及全纯映射。它们在我们通过数学手段理解自然界方面发挥了重要作用。它们在几何学和拓扑学中也有着广泛的应用，如Seiberg-Witten理论、Calabi-Yau空间的镜像对称性等。这些问题的解决将为一些物理理论提供数学基础，并对解决长期存在的数学问题有着深远的应用。大多数自然现象都是非线性的，具有奇异性。这些都反映在描述这些现象的微分方程可能的奇异解。对这些奇异解有一个完整的数学理解仍然是具有挑战性的。这项建议将解决其中一些基本问题。我也将尝试找到这些非线性方程的更多解-并将其应用于几何和拓扑学中的基本问题的研究。

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.