Pseudoholomorphic curves in low-dimensional topology

摘要奖：DMS-0806037主要研究者：Michael L. Hutchings该项目的主要部分是发展“嵌入接触拓扑学”（ECH），这是一种新的接触三维流形的不变量，它是根据Reeb流的周期轨道和辛化中的嵌入伪全纯曲线定义的。 ECH在结构上同构于Seiberg-Witten和Ozsvath-Szabo Floer同调的版本。 该项目的一些具体目标是进一步发展ECH的分析基础;创建用于计算ECH的工具，特别是在开卷分解方面;使用ECH获得Reeb轨道数量的下限;并努力将ECH扩展到一个更一般的理论，将其与Ozsvath-Szabo Floerhomology统一起来。 一些更广泛的目标是使用ECH机器来帮助计算三维辛场论，并探索族的Floer理论不变量。在这个项目中开发的嵌入接触同调位于动力学和低维拓扑之间的界面上。动力学关注物理系统随时间的行为，而低维拓扑研究三维和四维弯曲空间的可能形状。 嵌入式接触动力学允许人们从对动力学的理解中获得深层的拓扑信息;反过来，也可以从拓扑条件中获得重要的动力学信息，如稳定构型的存在性。

AbstractAward: DMS-0806037Principal Investigator: Michael L. HutchingsThe main part of the project is to develop "embedded contacthomology" (ECH), a new invariant of a contact three-manifold,which is defined in terms of periodic orbits of the Reeb flow andembedded pseudoholomorphic curves in the symplectization. ECH isconjecturally isomorphic to versions of the Seiberg-Witten andOzsvath-Szabo Floer homologies. Some specific goals of theproject are to further develop the analytical foundations of ECH;to create tools for computing ECH, particularly in terms of openbook decompositions; to use ECH to obtain lower bounds on numbersof Reeb orbits; and to work towards extending ECH to a moregeneral theory which would unify it with the Ozsvath-Szabo Floerhomology. Some broader goals are to use ECH machinery to helpcompute symplectic field theory in three dimensions, and toexplore Floer-theoretic invariants of families.The embedded contact homology developed in this project lies onthe interface between dynamics and low-dimensional topology.Dynamics is concerned with the behavior of physical systems overtime, while low-dimensional topology studies the possible shapesof curved spaces in three and four dimensions. Embedded contacthomology allows one to obtain deep topological information froman understanding of dynamics; and conversely to obtain importantdynamical information, such as the existence of stableconfigurations, from topological conditions.