Symplectic Topology and its Applications

建议：DMS-9971454首席研究员：陈伟民摘要：本项目将围绕辛拓扑和几何三个主题展开。在第一部分中，主要研究人员(与Y.Ruan联合)将建立辛奥尔布洛德的Gromov-Witten不变量。这项工作旨在通过将辛流形分解成两部分来计算其Gromov-Witten不变量。这样的运算一般都会引入轨道奇点，因此建立具有轨道奇点的辛流形的Gromov-Witten不变量将为计算Gromov-Witten不变量的程序奠定基础，该程序在辛拓扑以及二次几何等相关领域都有很大的潜在应用。其他动机来自理论物理的镜面对称性和弦理论，在这些理论中，需要考虑具有受控奇点的空间，如Ordiold奇点。在这个项目的第二部分，主要研究人员将继续他在三维Reeb动力系统上的工作，利用Seiberg-Witten Floer同调和接触同调之间的潜在联系。最后一部分是关于第二同伦群为零的光滑4-流形上的辛结构。目前，人们对它们的了解很少。近年来，在理论物理思想的输入下，人们目睹了几何和拓扑学领域不同数学分支之间的一些巨大互动。这个项目试图利用这些不同领域之间的密切相互作用来研究一些非常有趣的或基本的问题，如量子上同调，双曲几何，三维空间上Reeb动力学的周期轨道的存在性，以及辛4-流形。首席研究员认为，这个项目的成功结果将使上面列出的领域的知识有显著的进步，这些领域不仅在数学中很重要，在现实生活中的问题也很重要。例如，Reeb动力学的重要性如下例所示。卫星在太阳、行星和月球引力作用下的运动在数学上被描述为Reeb动力学。这个项目的相关部分旨在解决三维空间上Reeb动力系统的已有20年历史的Weinstein猜想，这是该领域最重要的猜想之一。

Proposal: DMS-9971454 Principal Investigator: Weimin ChenAbstract: This project will center on three topics in symplectic topology and geometry. In the first part the principal investigator (joint with Y. Ruan) will establish the Gromov-Witten invariants for symplectic orbifolds. This work aims at computing the Gromov-Witten invariants of a symplectic manifold by decomposing it into two pieces. Such an operation will in general introduce orbifold singularities, so the establishment of Gromov-Witten invariants for symplectic manifolds with orbifold singularities will lay the foundation for a program of computing Gromov-Witten invariants, which has great potential applications in symplectic topology as well as other related fields such as birational geometry. Other motivations come from mirror symmetry and string theory of theoretical physics, in which there is a demand to consider spaces with controlled singularities such as orbifold singularities. In the second part of this project the principal investigator will continue his work on the 3-dimensional Reeb dynamical systems by exploiting a potential connection between Seiberg-Witten Floer homology and the contact homology. The last part concerns symplectic structures on smooth 4-manifolds with vanishing second homotopy group. Currently very less is known about them. In recent years, one has witnessed some great interactions between different branches of mathematics in the area of geometry and topology, with the input of ideas from theoretical physics. This project seeks to exploit the intimate interplay between these different fields to investigate some very interesting or fundamental questions in such areas as quantum cohomology, birational geometry, the existence of periodic orbits of Reeb dynamics on a 3-dimensional space, and symplectic 4-manifolds. The principal investigator believes that a successful outcome of this project will make a significant advancement of knowledge in the areas listed above, which are important not only within mathematics but also in real life problems. For example, the importance of Reeb dynamics is seen in the following example. The motion of a satellite in the presence of the gravitational forces of the sun, the planets and the moon is described mathematically as a Reeb dynamics. The relevant part of this project aims at solving the 20-year-old Weinstein conjecture for Reeb dynamical systems on a 3-dimensinal space, which is one of the most important conjectures in the field.