Pseudoholomorphic Curves in Low-Dimensional Topology

DMS-0204681 Michael Hutchings该项目涉及低维、辛和接触拓扑学中新枚举不变量的开发和应用。 有两个主要目标。 第一个是发展和计算“周期Floer同调”，这是一个为保面积曲面自同构定义的理论，它计算周期轨道与R中嵌入的伪全纯曲线一起穿过映射环面。 这个理论被证明与映射环面的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。 此外，三维接触流形的周期Floer同调的一个类似物应该应用于过扭曲接触结构和光滑四维流形的拓扑。 第二个主要目标是开发和计算在不同版本的Floer理论中的等价对象族的不变量。 这些给出了辛同胚、三流形、勒让德结和任何其他类型的可以定义弗洛尔理论的对象的拓扑不变量。这个项目符合开发工具来理解三维和四维空间可能的整体形状的广泛主题。 例如，我们生活的宇宙是一个四维空间，如果其中包括时间的话，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。 一类重要的此类对象是伪全纯曲线，它们是类似肥皂膜的曲面。 通过计算空间中存在的具有适当约束的曲面的数量，可以获得关于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.