Symplectic Topology and its Applications

项目名称：DMS-9971454主要研究者：陈伟民摘要：本项目将围绕辛拓扑和几何中的三个主题展开。 在第一部分中，主要研究者（与Y.阮）将建立Gromov-Witten不变量的辛orbifolds。这项工作的目的是计算Gromov-Witten不变量的辛流形分解成两个部分。这样的操作一般会引入轨道奇点，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，该程序在辛拓扑学以及其他相关领域如双有理几何中具有巨大的潜在应用。 其他的动机来自镜像对称和理论物理学的弦理论，其中需要考虑具有受控奇点的空间，例如轨道奇点。在该项目的第二部分，主要研究人员将继续他的工作，通过利用Seiberg-Witten Floer同调和接触同调之间的潜在联系，对三维Reeb动力系统。最后一部分讨论了第二同伦群为零的光滑4-流形上的辛结构。目前对它们知之甚少。近年来，人们目睹了几何学和拓扑学领域不同数学分支之间的一些重大互动，并引入了理论物理学的思想。该项目旨在利用这些不同领域之间的密切相互作用，以研究一些非常有趣或基本的问题，如量子上同调，双有理几何，三维空间上Reeb动力学周期轨道的存在性，以及辛4流形。主要研究者认为，该项目的成功结果将使上述领域的知识取得重大进展，这些领域不仅在数学领域而且在真实的生活问题中都很重要。例如，Reeb动力学的重要性在以下示例中可见。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。本项目的相关部分旨在解决三维Reeb动力系统的Weinstein猜想，这是该领域最重要的成果之一。

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.