FRG: Holomorphic Curves in Low Dimensional Topology

DMS-0244663 Yakov Eliashberg，John Etnyre，Wu-Chung Hsiang，Michael Hutchings，Tomasz Mrowka，Peter Ozsvath，Ronald Stern，Zoltan Szab.这是数学科学分部重点研究组颁发的奖项，该奖项最近作为低维拓扑学的有力工具出现。本项目的目标是统一和协调全纯曲线应用领域的研究，以便：-构造新的3维和4维流形的不变量；-发展证明流形之间微分同胚的新方法；-建立新的方法来证明流形之间的微分同胚；-找到弦理论和辛几何之间的新关系，另一方面寻找纽结理论和三维流形的拓扑。这个项目的核心，嵌入辛场理论，有望为低维拓扑学中许多看似不同的问题提供一个统一的方法。这个项目的影响预计将比其直接目标发展低维拓扑学广泛得多。新发展的方法有望为我们理解拓扑学和理论物理，特别是弦理论之间的联系开辟新的天地。该项目也应该具有重要的教育价值。特别是，在这个项目下，将开发一个学位论文主题数据库，这是一个积极管理和支持的问题清单，将作为高级项目人员研究生博士学位论文主题的来源。

DMS-0244663Yakov Eliashberg, John Etnyre, Wu-Chung Hsiang, Michael Hutchings, Tomasz Mrowka, Peter Ozsvath, Ronald Stern, Zoltan Szabo.This is a Division of Mathematical Sciences Focused Research Group (FRG) award made under solicitation http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htmHolomorphic curves recently have emergedas a powerful tool in low-dimensional topology.The goal of this project is to unite and coordinateresearch in the area of applications of holomorphiccurves in order to: - construct new invariants of manifolds of dimension 3 and 4;- develop new methods for proving diffeomorphism between the manifolds;- find new relations between string theory and symplectic geometry on the one side, and knot theory and topology of 3-manifolds on the other.The core of this project, Embedded SymplecticField Theory, is expected to provide a unified approachto many seemingly different problems in low-dimensionaltopologyThe impact of this project is expected to be much broaderthan its immediate goal of developing low-dimensionaltopology. There is a hope that the newly developed methodsopen new horizons in our understanding of linksbetween Topology and Theoretical Physics, and in particularString Theory. The project should also have a significanteducational value. In particular, under this project therewill be developed a Dissertation Subject Database,an actively managed and supported list of problems which willserve as a source of topics of PhD dissertation for graduatestudents of the senior project personnel.