Pseudoholomorphic Curves in Low-Dimensional Topology

DMS-0204681 Michael Hutchings这个项目涉及低维、辛拓扑、辛拓扑和接触拓扑学中新的枚举不变量的开发和应用。有两个主要目标。第一种是发展和计算“周期Floer同调”，这是一种定义为保面积曲面微分同态的理论，它计算与R中嵌入的伪全纯曲线一起穿过映射环面的周期轨道。这一理论被推测为与映射环面的Seiberg-WittenFloer同调一致，从而给出了低维拓扑和表面动力学之间的联系。同时，三维接触流形的周期Floer同调的一个类比也应该应用于过扭接触结构和光滑四维流形的拓扑。第二个主要目标是发展和计算不同版本的Floer理论中等价物体族的不变量。它们给出了辛同构、三维流形、勒让德结和任何其他类型的物体的拓扑不变量，对于任何其他类型的物体，可以定义一种版本的Floer理论。这个项目符合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，我们生活的宇宙是一个四维空间，如果包括时间，它的全局结构是未知的。这里使用的工具来理解空间的形状，涉及到空间中有趣的几何对象。这类物体的一个重要类别是伪全纯曲线，它们的表面类似于肥皂膜。通过计算空间中存在的具有适当约束的此类曲面的数量，可以获得关于该空间的全局结构的信息。

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.