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Pin(2)-Symmetry in Monopole Floer Homology

单极Floer同调中的Pin(2)-对称性

基本信息

批准号：
1948820
负责人：
Francesco Lin
金额：
$ 12.38万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1948820&HistoricalAwards=false
关键词：
Pin Symmetry Monopole Floer Homology

项目摘要

The main focus of this National Science Foundation funded project is the interaction between two areas of research known as gauge theory and low-dimensional topology. The former is the language in which many physical theories are formulated, while the latter studies the shapes of three- and four-dimensional spaces. It is common to apply knowledge of geometry to predict the behavior of physical systems. For example, it is straightforward to predict the sound of a drum from its shape. The opposite approach is sometimes also feasible; in particular, the study of differential equations originating from gauge theories can lead to very deep insights about the topology of spaces. In recent ground-breaking work, Ciprian Manolescu disproved the almost hundred-year-old Triangulation conjecture, which roughly asserted that in higher dimensions, every space can be cut into very simple pieces. The main goal of this project is to use the PI's recently developed computational techniques to understand in greater detail several properties of the object studied by Manolescu, called the three-dimensional homology cobordism group.This project has two main goals. First, to explore the properties of Pin(2)-monopole Floer homology, a package of invariants of three-manifolds defined in analogy to Manolescu's construction within the Morse-theoretic framework of Kronheimer and Mrowka. The Pin(2)-symmetry is reflected in an extremely rich A-infinity structure on these invariants, and we will focus on understanding the higher operations and their implications for natural topological operations such as connected sums. Second, to apply these tools to study problems in low-dimensional topology, focusing particularly on the homology cobordism group in dimension three. Very little is known about this group, especially regarding the existence of torsion, and Pin(2)-monopole Floer homology may provide an avenue to solve many of these problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个国家科学基金会资助的项目的主要重点是两个研究领域之间的相互作用，称为规范理论和低维拓扑结构。前者是许多物理理论的语言，而后者研究三维和四维空间的形状。应用几何知识来预测物理系统的行为是很常见的。例如，从鼓的形状来预测它的声音是很简单的。相反的方法有时也是可行的;特别是，对源于规范理论的微分方程的研究可以导致对空间拓扑的非常深刻的见解。在最近的开创性工作中，Ciprian Manolescu反驳了近百年历史的三角测量猜想，该猜想大致断言，在更高的维度中，每个空间都可以切割成非常简单的碎片。该项目的主要目标是使用PI最近开发的计算技术来更详细地了解Manolescu研究的对象的几个属性，称为三维同源配边群。首先，研究Pin（2）-M-Floer同调的性质，Pin（2）-M-Floer同调是在Kronheimer和Mrowka的Morse理论框架内，类似于Manolescu的构造而定义的一组三流形的不变量。Pin（2）对称性反映在这些不变量上极其丰富的A-无穷结构中，我们将专注于理解更高的运算及其对自然拓扑运算（如连通和）的影响。第二，应用这些工具研究低维拓扑中的问题，特别是三维空间中的同调配边群。关于这一群的研究知之甚少，特别是关于扭转的存在，而Pin（2）-Nickel Floer同源性可能为解决这些问题提供了一条途径。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。