Pseudoholomorphic curves in low-dimensional topology

摘要奖：DMS-0806037首席研究员：Michael L.Hutchings该项目的主要部分是开发一种新的接触三流形的不变量，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猜想同构于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；创建计算ECH的工具，特别是在Openbook分解方面；使用ECH获得Reeb轨道数的下界；以及致力于将ECH扩展为更一般的理论，将其与Ozsvath-Szabo Floer同调统一。一些更广泛的目标是使用ECH机制来帮助计算三维辛场理论，并探索族的Floer理论不变量。本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。动力学涉及物理系统的超时行为，而低维拓扑研究三维和四维弯曲空间的可能形状。嵌入式接触学允许人们从对动力学的理解中获得深刻的拓扑信息，反之，从拓扑条件中获得重要的动力学信息，如稳定构型的存在。

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.