Pseudoholomorphic curves in low-dimensional topology

本项目的目标是在Floer同调的框架下发展新的拓扑不变量。 该项目的一个主要部分是开发“嵌入接触同调”，一个新的不变量的contactthree-manifold计数嵌入pseudoholomorphic曲线在四维辛的三维流形。 嵌入接触同调在结构上同构于Seiberg-Witten或Ozsvath-Szabo Floer同调的一个版本。 它提供了三维和四维光滑流形拓扑与全纯曲线和接触结构的几何和动力学之间的桥梁。 该项目的另一部分是为不同版本的Floer理论构建等价对象族的Floer理论不变量，从而获得三流形族、辛同胚、Legendrian结和其他类型的Floer理论可以定义的对象的同伦不变量。该项目符合开发工具以理解三维和四维空间可能的整体形状的广泛主题，比如我们生活的宇宙。 这里用来理解空间形状的工具包括计算空间内有趣的几何物体。 其中一类重要的对象是伪全纯曲线，它是一种类似肥皂膜的曲面，通过计算空间中存在的此类曲面的数量，可以获得关于空间整体结构的深层信息。

The goal of this project is to develop new topological invariants inthe framework of Floer homology. A major part of the project is todevelop "embedded contact homology", a new invariant of contactthree-manifolds which counts embedded pseudoholomorphic curves in thefour-dimensional symplectization of the three-manifold. Embeddedcontact homology is conjecturally isomorphic to a version of theSeiberg-Witten or Ozsvath-Szabo Floer homologies. It provides abridge between the topology of smooth manifolds in three and fourdimensions, and the geometry and dynamics of holomorphic curves andcontact structures. Another part of the project is to constructFloer-theoretic invariants of families of equivalent objects fordifferent versions of Floer theory, thus obtaining homotopy invariantsof families of three-manifolds, symplectomorphisms, Legendrian knots,and other types of objects for which Floer theory can be defined.This project fits into the broad theme of developing tools tounderstand the possible global shapes of three- and four-dimensionalspaces, such as the universe that we live in. The tools used here tounderstand the shape of a space involve counting interesting geometricobjects inside the space. An important class of such objects arepseudoholomorphic curves, which are surfaces resembling soap films.By counting the number of such surfaces that exist in a space, one cangain deep information about the global structure of the space.