Heegaard Floer homology, concordance, and categorification

首席研究员将使用Heegaard flower同源学的工具来研究低维拓扑中的现象。一个目标是更好地理解结和谐组，特别是平滑和拓扑类别之间的差异。来自弗洛尔同调的一致性不变量，如d不变量和ε不变量，非常适合这项任务。具体而言，主要研究者旨在证明由拓扑切片结生成的光滑协调群的子群包含无限秩和。一种不同的一致性方法是通过边界花同源性。在同样的意义上，结花同调对亚历山大多项式进行了分类，人们可以希望Seifert曲面的有边花同调对Seifert形式进行了分类。许多经典的一致性不变量可以用Seifert形式来定义，该项目将研究这些不变量如何在有边界的不变量中表现出来。空间中打结曲线的结构与我们对三维和四维空间形状的理解密切相关。将时间视为第四维，结的一致性研究就变成了一个关于结随时间演化的问题。通过对结的演化方式施加不同的限制，人们可以得到结复杂性的不同定义。这些复杂性测量方法的应用范围从非常小的（例如，DNA链结的行为）到非常大的（例如，宇宙的形状）。

The principal investigator will use tools from Heegaard Floer homology to study phenomena in low-dimensional topology. One goal is to better understand the knot concordance group, in particular the difference between the smooth and topological categories. Concordance invariants coming from Floer homology, such as d-invariants and epsilon, are well-suited for this task. Specifically, the principal investigator aims to show that the subgroup of the smooth concordance group generated by topologically slice knots contains an infinite rank summand. A different approach to concordance is via bordered Floer homology. In the same sense that knot Floer homology categorifies the Alexander polynomial, one may hope that the bordered Floer homology of a Seifert surface categorifies the Seifert form. Many classical concordance invariants can be defined in terms of the Seifert form, and the project will investigate how these invariants manifest themselves within the bordered invariants.The structure of knotted curves in space is intimately related to our understanding of the shape of 3- and 4-dimensional space. Thinking of time as the fourth dimension, the study of knot concordance becomes a question about the evolution of knots over time. By putting different restrictions on how knots may evolve, one obtains different definitions of the complexity of a knot. These measures of complexity have applications from the very small (e.g., the behavior of knotted strands of DNA) to the very large (e.g., the shape of the universe).