Pseudoholomorphic curves in low-dimensional topology

该项目的目标是在Floer同调框架内开发新的拓扑不变量。 该项目的一个主要部分是开发“嵌入接触同调”，这是一种新的接触三流形不变量，它在三流形的四维辛化中计算嵌入的伪全纯曲线。 据推测，嵌入接触同源性与 Seiberg-Witten 或 Ozsvath-Szabo Floer 同源性的版本同构。 它在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架起了一座桥梁。 该项目的另一部分是为不同版本的 Floer 理论构造等效对象族的 Floer 理论不变量，从而获得三流形族、辛同胚、勒让结以及可以定义 Floer 理论的其他类型对象的同伦不变量。该项目符合开发工具的广泛主题，以了解三维和四维空间（例如我们生活的宇宙）可能的全局形状。要了解空间的形状，需要计算空间内有趣的几何对象。 一类重要的此类物体是伪全纯曲线，它们是类似于肥皂膜的表面。通过计算空间中存在的此类表面的数量，人们可以获得有关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.