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纽结协调群，特别是光滑范畴和拓扑范畴之间的区别。来自Floer同调的调和不变量，如d-不变量和epsilon，非常适合于这一任务。具体地说，主要研究人员的目的是证明由拓扑片纽结生成的光滑协调群的子群包含无限秩和。另一种不同的方法是通过有边界的Floer同调。在相同的意义上，纽结Floer同调归类于Alexander多项式，人们可能希望Seifert曲面的有界Floer同调归类为Seifert形式。许多经典的调和不变量可以用Seifert形式来定义，本项目将研究这些不变量是如何在有界不变量中表现出来的。空间中纽结曲线的结构与我们对三维和四维空间的形状的理解密切相关。将时间视为第四维，研究结的一致性就成了一个关于结随时间演变的问题。通过对结的进化方式施加不同的限制，可以得到对结的复杂性的不同定义。这些复杂性的测量从非常小的(例如，打结的DNA链的行为)到非常大的(例如，宇宙的形状)都有应用。

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).