Pin(2)-Symmetry in Monopole Floer Homology

这个由美国国家科学基金会资助的项目的主要焦点是两个研究领域之间的相互作用，即规范理论和低维拓扑。前者是表述许多物理理论的语言，而后者研究三维和四维空间的形状。应用几何学知识来预测物理系统的行为是很常见的。例如，从鼓的形状来预测鼓的声音是很直接的。相反的方法有时也是可行的；特别是，对源自规范理论的微分方程的研究可以对空间的拓扑结构产生非常深刻的见解。在最近的突破性工作中，Ciprian Manolescu推翻了将近百年的三角测量猜想，该猜想粗略地断言，在更高的维度中，每个空间都可以被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所研究的对象的几个特性，称为三维同调协群。这个项目有两个主要目标。首先，在Kronheimer和Mrowka的morse理论框架内，利用与Manolescu构造相似的一组三流形不变量，探索Pin(2)-单极子Floer同调的性质。Pin(2)-对称性反映在这些不变量上极其丰富的a -∞结构中，我们将重点理解更高的运算及其对自然拓扑运算（如连通和）的含义。其次，将这些工具应用于低维拓扑问题的研究，特别是在三维同调协群上。我们对这个群知之甚少，特别是关于扭转的存在，而Pin(2)-单极子Floer同源性可能为解决这些问题提供了一条途径。该奖项反映了美国国家科学基金会的法定使命，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，被认为值得支持。

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.