Heegaard Floer homology, concordance, and categorification

首席研究员将使用Heegaard Floer同调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一致性组，特别是光滑和拓扑类别之间的差异。来自Floer同源性的一致性不变量，如d-不变量和coherent，非常适合这项任务。具体而言，主要研究者的目的是表明，拓扑切片结所产生的光滑和谐组的子群包含一个无限秩和。另一种不同的方法是通过边界Floer同源性。在纽结弗洛尔同调将亚历山大多项式归类的相同意义上，人们可能希望塞弗特曲面的加边弗洛尔同调将塞弗特形式归类。许多经典的和谐不变量可以用塞弗特形式定义，该项目将研究这些不变量如何在边界不变量中表现出来。空间中的纽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。把时间看作第四维，纽结和谐的研究就变成了一个关于纽结随时间演变的问题。通过对纽结如何演化施加不同的限制，人们得到了对纽结复杂性的不同定义。这些复杂性的度量从非常小的（例如，DNA的打结链的行为）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).