Topological Applications of Floer Theory

规范理论是对一种特殊的非线性微分方程的研究，它最初出现在量子物理学中。在数学上，这些方程的解可以用来理解底层空间的拓扑(形状)。在研究三维形状时，来自规范理论的信息被编码到称为Floer同调的代数结构中。该项目旨在研究Floer同调的不同版本及其在拓扑学中的应用。特别是，尽管Floer同调与三维空间有关，但通过间接的途径，它可以给出五维或更高维空间的三角剖分的见解；对这些三角剖分进行分类是一个主要的开放问题。三角剖分是将空间分解成多面体，并给出空间的简单组合描述。该项目还旨在从量子物理学的最新进展中获得灵感，构建Floer同源的新变体。另一方面，通过与计算机科学家的合作，提出将拓扑学应用于分布式计算模型的分类。该项目涉及Floer理论及其在低维和高维流形研究中的应用。特别是，PI将研究三维流形的Seiberg-Witten Floer稳定同伦型以及相关的Pin(2)-等变Seiberg-Witten Floer同调。这些理论可以用来获得关于三维同调余边群的信息。反过来，同调共边给出了至少五维流形的三角剖分的分类。类似的方法将被用来研究自旋四维流形的相交形式。PI还将致力于构建Heegaard Floer同调和纽结Floer同调的Pin(2)版本，并开发Heegaard Floer同调的新计算技术。此外，在与计算机科学家的持续合作中，PI正在探索拓扑学在分布式计算中的应用；在那里，由几台计算机组成的系统对给定任务的可解性可以用同伦理论中的一个问题来重新表述。

Gauge theory is the study of a particular kind of non-linear differential equations, that originally appeared in quantum physics. In mathematics, the solutions to these equations can be used to understand the topology (the shape) of the underlying space. When studying three-dimensional shapes, the information from gauge theory is encoded into an algebraic structure called Floer homology. The project aims to study different versions of Floer homology and their applications to topology. In particular, although Floer homology is associated to three-dimensional spaces, by an indirect route it can give insights into the triangulations of spaces of dimension five or higher; classifying these triangulations is a major open problem. A triangulation is a decomposition of the space into polyhedra, and gives a simple combinatorial description of the space. The project also aims at constructing new variants of Floer homology, with inspiration drawn from recent advances in quantum physics. In a different direction, through collaboration with computer scientists, it is proposed to apply topology to the classification of distributed computing models.The project concerns Floer theory and its applications to the study of both low dimensional and high dimensional manifolds. In particular, the PI will investigate the Seiberg-Witten Floer stable homotopy types of three-manifolds and the associated Pin(2)- equivariant Seiberg-Witten Floer homology. These theories can be used to get information about the homology cobordism group in three dimensions. In turn, homology cobordism gives insight into the classification of triangulations for manifolds of dimension at least five. Similar methods will be used to study the intersection forms of spin four-manifolds. The PI will also work on constructing Pin(2) versions of Heegaard Floer homology and knot Floer homology, and on the development of new computational techniques for Heegaard Floer homology. Furthermore, in an ongoing collaboration with computer scientists, the PI is exploring applications of topology to distributed computing; there, the solvability of a given task by a system of several computers can be rephrased in terms of a question in homotopy theory.