Pseudoholomorphic curves in low-dimensional topology

这个项目的目标是在Floer同调的框架下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维流形的不变量，它将嵌入的伪全纯曲线计算在三维流形的四维辛化中。嵌入接触同调猜想同构于Seiberg-Witten或Ozsvath-Szabo Floer同调的一种形式。它提供了三维和四维光滑流形的拓扑与全纯曲线和接触结构的几何和动力学之间的桥梁。该项目的另一个部分是为不同版本的Floer理论构造等价对象族的Floer理论不变量，从而获得三维流形、辛同态、勒让德结和其他类型的可以定义Floer理论的对象族的同伦不变量。这个项目适合于开发工具的广泛主题，以了解三维和四维空间可能的全局形状，例如我们生活的宇宙。这里使用的工具来理解空间的形状，包括计算空间中有趣的几何对象。伪全纯曲线是一类重要的空间对象，它是一种类似于肥皂膜的表面，通过计算空间中存在的这种表面的数目，可以获得关于空间全局结构的更深层次的信息。

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.