FRG: Collaborative Research: Floer Homotopy Theory

FRG：合作研究：弗洛尔同伦理论

• 批准号: 1563615
• 负责人: Ciprian Manolescu
• 金额: $ 36.51万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1563615&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Floer; Homotopy

Topology is the study of those properties of shapes that are unchanged by stretching and bending. Over the years, mathematicians have developed various topological invariants, or in other words, quantities that are associated to shapes and can distinguish between those that have different properties. Homology is a well-known such invariant, which can be associated to any multi-dimensional shape, and which is a quantitative measure of the number of holes in a space. A circle, for instance, has only a one-dimensional hole, whereas the surface of a doughnut has two one-dimensional holes, a meridian and a longitude, that are not filled in by the surface itself, and an additional two-dimensional hole. Floer homology is a more refined invariant that is responsible for some of the most important recent advances in the study of knotted closed loops in space, three-dimensional shapes, and shapes with a geometry known as a symplectic structure that is exhibited by phase spaces in classical mechanics. This project brings together several researchers working in different areas of topology and geometry to study Floer homology. The main goal of the project is the following: To every knot, three-dimensional shape, or symplectic shape, one should associate a different object, called a Floer space or a Floer homotopy type, whose (ordinary) homology is the Floer homology of the initial shape. This has been accomplished so far in a limited number of cases. A general theory of Floer spaces will lead to new advances in several areas. Furthermore, the study of Floer spaces will be based on techniques from a subfield of topology called homotopy theory. This project will create a community of scholars at the interface of these current and extremely research active areas of mathematics.Floer homology is a fundamental tool in geometry and topology, whose applications range from the Arnold conjecture to the surgery characterization of various knots. Floer homology has also laid the basis for completely unexpected interconnections between algebraic and symplectic geometry in the form of homological mirror symmetry. Floer homotopy theory, an extension to spaces rather than homology groups, has been implemented in a small number of cases, leading to significant applications, for example, the resolution of the triangulation conjecture in high dimensions and work on immersed Lagrangian spheres. Further, the ideas behind Floer homotopy inspired the construction of a Khovanov homotopy type associated to knots in the three-sphere. The main scientific goal of this project is to give a general construction of Floer homotopy. The necessary foundational work will build upon recent advances in multiple areas. These include the conceptual advances in equivariant stable homotopy theory stemming from the resolution of Kervaire invariant one problem, and the development of new approaches to define virtual fundamental classes in Floer theory. The project aims to put the homotopical and homological variants of Floer theory on equal footing. As a consequence, new applications in both symplectic and low-dimensional topology are anticipated, for example: (i) a spectral Fukaya category associated to a symplectic manifold will be constructed; (ii) the Heegaard Floer theory of Ozsvath and Szabo will be used to produce a computable invariant parallel to the celebrated Bauer-Furuta invariant for four-manifolds; (iii) Seiberg-Witten Floer homotopy types will be studied using the tools of equivariant stable homotopy theory; and (iv) the Khovanov homotopy type will be extended to give invariants of knot cobordisms and tangles.

拓扑是对伸展和弯曲不变的形状特性的研究。多年来，数学家已经开发了各种与形状相关的数量，并且可以区分具有不同特性的拓扑数量。同源性是一种众所周知的这种不变，可以与任何多维形状相关联，并且是对空间中孔数量的定量度量。例如，一个圆只有一个维孔，而甜甜圈的表面具有两个一维孔，一个子午线和一个经度，它们不会被表面本身填充，还有一个额外的二维孔。 Floer同源性是一种更精致的不变性，负责在空间中打结的封闭环，三维形状和形状的一些最新进展，其形状具有几何形状，称为符号结构，在古典力学中被相位空间表现​​出来。该项目汇集了一些在拓扑和几何学不同领域工作的研究人员，以研究浮动同源性。该项目的主要目标是：对于每个结，三维形状或符号形状的每个结，一个人应将一个不同的物体关联，称为漂浮空间或浮动同拷贝类型，其（普通的）同源性是初始形状的浮动同源性。到目前为止，在有限数量的情况下，这已经完成。浮空间的一般理论将导致多个领域的新进展。此外，对浮空间的研究将基于来自拓扑统一的统一理论的技术。该项目将在这些当前和非常研究的数学领域的界面上创建一个学者社区。Floer同源性是几何和拓扑的基本工具，其应用从Arnold的猜想到各种结的手术表征。漂浮物同源性还为以同源镜子对称形式的代数和符号几何形状之间完全出乎意料的互连奠定了基础。在少数情况下，已经实施了浮点同质理论，是对空间而不是同源组的扩展，导致了重要的应用，例如，在高维度中解决三角剖分的猜想，并在浸入浸入的拉格朗日球体上进行工作。此外，Floer同拷贝背后的想法启发了与三球结中的结相关的Khovanov同型类型的构建。该项目的主要科学目标是提供一般的浮子同质构造。必要的基础工作将基于多个领域的最新进展。其中包括源自kervaire不变的一个问题的概念稳定同质理论的概念进步，以及开发了定义浮点理论中虚拟基本类别的新方法。该项目旨在将浮子理论的同源和同源变体保持平等地位。结果，预计将在符号和低维拓扑的新应用中进行新的应用，例如：（i）将构建与符号歧管相关的光谱福卡亚类别； （ii）Ozsvath和Szabo的Heegaard Floer理论将用于产生与四个manifolds的著名鲍尔 - 毛图尔不变的可计算不变的； （iii）将使用均衡稳定同质理论的工具研究Seiberg-witten的浮子同质类型； （iv）Khovanov同型类型将扩展，以产生结的恢复和缠结。