Nonlinear Problems in Symplectic Geometry and Complex Geometry

摘要建议：DMS-9802479主要研究人员：冈田该项目解决了辛几何和Kaehler几何中的一些基本问题。第一部分是关于具有正标量曲率的Kaehler-Einstein度规的存在性。这些度量提供了黎曼流形上爱因斯坦方程的解。在第二部分，主要研究者(PI)打算进一步研究一般辛流形的新的不变量，通常被称为GW不变量，并在哈密顿系统、辛拓扑和几何中找到更多的应用。这一部分还包含了理解GW不变量的结构及其与特定可积系统的可能关系的非常基本的问题。PI还建议研究一些与物理和几何有关的非线性偏微分方程组问题。本项目的问题是出于我们希望在合适的条件下探索基本物理定律的数学的愿望。例如，GW不变量受到数学物理中结合重力的Sigma模型理论的启发，它们扩展了经典的计数几何，该几何涉及通过许多点或曲线的交点对曲线进行计数。这些问题的解决将对广义相对论中的爱因斯坦方程、量子场论中的爱因斯坦方程、弦理论中的镜像对称现象提供新的数学见解。我们的研究也将加深我们对辛流形、代数流形和Calabi-Yau空间性质的理解。

AbstractProposal: DMS-9802479Principal Investigator: Gang TianThe project addresses some fundamental problems from symplectic geometry and Kaehler geometry. The first part concerns the existence of Kaehler-Einstein metrics with positive scalar curvature. These metrics provide solutions of the Einstein equation on Riemannian manifolds. In the second part, the principal investigator (PI) intends to further study new invariants, often refered as GW-invariants, of general symplectic manifolds and find more applications to Hamiltonian systems, symplectic topology and geometry. This part also contains very basic problem of understanding structure of GW-invariants and its possible relation to particular integrable systems. The PI also suggests to study some related nonlinear PDE problems which arise from physics and geometry.Problems in this project were motivated by our desire of probing mathematics of basic physical laws under suitable conditions. For example, GW-invariants were inspired by a sigma model theory coupled with gravity in mathematical physics, and they extend classical enumerative geometry which involves counting curves through a number of points or intersections of curves. The resolutions of these problems will provide new mathematical insights of Einstein equation in general relativity, quantum field theory, mirror symmetry phenomenon in string theory. Our study will also deepen our understanding properties of symplectic manifolds, algebraic manifolds and Calabi-Yau spaces.