Pseudoholomorphic Curves in Low-Dimensional Topology

michael hutchings这个项目涉及低维、辛和接触拓扑中新的枚举不变量的发展和应用。有两个主要目标。首先，发展并计算了“周期花同调”理论，这是一个定义于保面积表面微分同调的理论，它将周期轨道与嵌入在映射环面上的伪全纯曲线一起计数。据推测，该理论与映射环面的Seiberg-WittenFloer同调一致，从而在低维拓扑和表面动力学之间建立了联系。此外，三维接触流形的周期flower同调的模拟应该应用于超扭曲接触结构和光滑四流形的拓扑结构。第二个主要目标是在不同版本的花理论中发展和计算等价对象族的不变量。这些给出了辛形态族的拓扑不变量，三流形，Legendrian结，以及任何其他类型的对象，对于这些对象，花理论的一个版本可以被定义。这个项目符合开发工具来理解三维和四维空间可能的全局形状的广泛主题。例如，我们生活的宇宙是一个四维空间，如果包括时间，它的整体结构是未知的。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。这类物体的一个重要类别是伪全纯曲线，它是类似于肥皂膜的表面。通过计算空间中存在的具有适当约束的曲面的数量，可以获得有关空间整体结构的信息。

DMS-0204681Michael HutchingsThis project involves the development and application of newenumerative invariants in low-dimensional, symplectic, and contacttopology. There are two main goals. The first is to develop andcompute ``periodic Floer homology'', a theory defined for anarea-preserving surface diffeomorphism, which counts periodic orbitstogether with embedded pseudoholomorphic curves in R cross the mappingtorus. This theory is conjectured to agree with the Seiberg-WittenFloer homology of the mapping torus, thus giving a link between lowdimensional topology and surface dynamics. Also, an analogue ofperiodic Floer homology for three-dimensional contact manifolds shouldhave applications to the topology of overtwisted contact structuresand smooth four-manifolds. The second main goal is to develop andcompute invariants of families of equivalent objects in differentversions of Floer theory. These give topological invariants offamilies of symplectomorphisms, three-manifolds, Legendrian knots, andany other type of object for which a version of Floer theory can bedefined.This project fits into the broad theme of developing tools tounderstand the possible global shapes of three and four dimensionalspaces. For example the universe we live in is a four dimensionalspace if one includes time, and its global structure is not known.The tools used here to understand the shape of a space involvecounting interesting geometric objects inside the space. An importantclass of such objects are pseudoholomorphic curves, which are surfacesresembling soap films. By counting the number of such surfaces withappropriate constraints that exist in a space, one can gaininformation about the global structure of the space.