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

Daniel Waite将使用弗洛尔理论和四维拓扑中的工具来解决各种嵌入和协同问题。20世纪80年代，唐纳森彻底改变了我们对光滑四维流形的理解，他表明微分方程的解空间来自理论物理，产生了新的拓扑不变量。在20世纪90年代，Seiberg和Witten给出了一组不同的方程，以物理学中的对偶性为指导，这些方程更容易处理，并提供了与唐纳森不变量大致相同的信息。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和结理论——三维空间中封闭曲线的研究。这主要得益于Oszvath和Szabo提出的Heegaard flower理论；这给出了一个不变量包，它（通过设计）是Seiberg-Witten不变量的重新表述，但更适合在三维中使用，更易于计算。Daniel将利用各种和谐和理性同调协不变量，结合各种四维结构，在两类问题上得到新的结果：1)给定三维球面上的一个结，它在四维球面上可以束缚什么样的嵌入面？2)给定一个三维流形，它可以绑定什么样的四维流形，它可以嵌入到什么样的四维流形中？这些问题之间有各种各样的关系，所以任何一个问题的进展都会导致另一个问题的进展。这项研究将引起英国和世界范围内从事规范理论和低维拓扑研究的数学家的极大兴趣。上面的问题1)与研究结之间的戈迪安距离密切相关，这是研究结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.