Pseudoholomorphic Curves in Low-Dimensional Topology

低维拓扑中的伪全纯曲线

• 批准号: 0204681
• 负责人: Michael Hutchings
• 金额: $ 8.28万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204681&HistoricalAwards=false
• 关键词: Pseudoholomorphic; Curves; Low; Dimensional; Topology

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.

DMS-0204681Michael Hutchings 该项目涉及低维、辛和接触拓扑中新的枚举不变量的开发和应用。 有两个主要目标。 第一个是开发和计算“周期性弗洛尔同调”，这是一种为保面积表面微分同胚定义的理论，它通过 R 中穿过映射环的嵌入伪全纯曲线来计算周期性轨道。 据推测，该理论与映射环面的 Seiberg-WittenFloer 同源性一致，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弗洛尔同调的类似物应该适用于过度扭曲接触结构和光滑四流形的拓扑。 第二个主要目标是开发和计算不同版本的弗洛尔理论中等效对象族的不变量。 这些给出了辛同态、三流形、勒让结以及可以定义弗洛尔理论版本的任何其他类型对象的拓扑不变量。该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，我们生活的宇宙如果包含时间，就是一个四维空间，并且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对象。 此类物体的一个重要类别是伪全纯曲线，它们是类似于肥皂膜的表面。 通过计算空间中存在的具有适当约束的此类表面的数量，可以获得有关空间整体结构的信息。