Low-dimensional Topology: Khovanov Homology and the Link with Heedaard-Floer Homology

Daniel Waite将使用Floer理论和四维拓扑学的工具来解决各种嵌入和协边问题。上世纪80年代，Donaldson彻底改变了我们对光滑4维流形的理解，他证明了来自理论物理的微分方程解空间产生了新的拓扑不变量。在20世纪90年代，Seiberg和Witten在物理学对偶性的指导下，给出了一组不同的方程，它们更容易处理，并提供了与Donaldson不变量或多或少相同的信息。在21世纪，这些不变量和相关的不变量已被应用于三维流形的研究以及纽结理论--研究三维空间中的闭合曲线。这主要得益于奥斯瓦特和萨博对希加·弗洛尔理论的发展；这给出了一组不变量，它们(经过设计)是Seiberg-Witten不变量的重新表述，但它们更适合于在3维中使用，并且更易于计算。Daniel将使用各种协调和有理同调协整不变量，以及各种4维构造，在两类问题中获得新的结果：1)给定3-球面中的一个结，它可以约束4维球中的哪种嵌入曲面？2)给定3维流形，它可以约束什么类型的4维流形，它可以嵌入什么4维流形？这些问题之间有各种关系，这项研究将对英国和世界各地从事规范理论和低维拓扑学工作的其他数学家产生相当大的兴趣。上面的问题1)与纽结之间的Gordian距离的研究密切相关，这是研究纽结DNA分子的数学生物学家感兴趣的。

Daniel Waite will use tools from Floer theory and 4-dimensional topology to attack various embedding and cobordism problems.Our understanding of smooth 4-dimensional manifolds was revolutionised in the 1980s by Donaldson, who showed that solution spaces to differential equations coming from theoretical physics gave rise to new topological invariants. In the 1990s Seiberg and Witten gave a different set of equations, guided by dualities in physics, which were easier to work with and give more or less the same information as Donaldson invariants. In the 21st century these and related invariants have been applied to the study of 3-dimensional manifolds and also to knot theory -- the study of closed curves in 3-dimensional space. This has primarily been facilitated by the development of Heegaard Floer theory by Oszvath and Szabo; this gives a package of invariants which (by design) are a reformulation of Seiberg-Witten invariants, but which are more adapted to use in 3 dimensions and are more amenable to calculation.Daniel will use various concordance and rational homology cobordism invariants, together with various 4-dimensional constructions, to obtain new results in two kinds of problem:1) given a knot in the 3-sphere, what kind of embedded surfaces in the 4-dimensional ball can it bound?2) given a 3-dimensional manifold, what kind of 4-dimensional manifolds can it bound, and what 4-manifolds can it be embedded into?There are various relations between these problems, so progress in either one can lead to progress in the other.This research will be of considerable interest to other mathematicians in the UK and worldwide working in gauge theory and low-dimensional topology. Problem 1) above is closely related to the study of Gordian distance between knots which is of interest to mathematical biologists studying knotted DNA molecules.