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

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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.