Pin(2)-Symmetry in Monopole Floer Homology

单极Floer同调中的Pin(2)-对称性

• 批准号: 1807242
• 负责人: Francesco Lin
• 金额: $ 16.79万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2019-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1807242&HistoricalAwards=false
• 关键词: Pin; Symmetry; Monopole; Floer; Homology

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的相互作用，即量规理论和低维拓扑。前者是制定许多物理理论的语言，而后者研究了三维和四维空间的形状。使用几何学知识来预测物理系统的行为是很常见的。例如，从其形状中预测鼓的声音是很简单的。相反的方法有时也是可行的。特别是，对源自量规理论的微分方程的研究可能会导致有关空间拓扑的非常深刻的见解。在最近的开创性作品中，Ciprian Manolescu反驳了几乎一百年历史的三角猜想，这大致断言在较高的维度上，每个空间都可以切成非常简单的零件。该项目的主要目的是使用PI最近开发的计算技术来详细了解Manolescu研究的对象的几种属性，称为三维同源性COBORDISM小组。该项目具有两个主要目标。首先，要探索PIN（2） - 单孔浮子同源性的性能，这是在Kronheimer和Mrowka的Morse理论框架内类似于Manolescu的构造中定义的三个manifolds的包装。 PIN（2） - 对称性反映在这些不变的非常丰富的A-内结构中，我们将专注于理解更高的操作及其对自然拓扑操作（例如连接的总和）的影响。其次，将这些工具应用于研究低维拓扑的问题，尤其是第三维中的同源性共生组。关于这一组的知之甚少，尤其是关于扭转的存在，PIN（2） - 单孔浮子同源性可能会提供解决许多问题的途径。该奖项反映了NSF的法定任务，并被认为值得通过评估获得支持。利用基金会的知识分子和更广泛的影响审查标准。