University College London, 03/11/2016
“Instantons and the four colour problem”
Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional Z/2 vector space. The main result about this instanton homology of graphs is a non-vanishing theorem, proved using techniques from 3-dimensional topology: if the graph is bridgeless, then its instanton homology is non-zero. It is not unreasonable to conjecture that, if the graph lies in the plane, then the dimension of its instanton homology is equal to the number of Tait colorings of the graph (essentially the same as four-colorings of the planar regions that the planar graph defines). If the conjecture were to hold, then the non-vanishing theorem for instanton homology would imply the four-color theorem and would provide a human-readable proof. There is some evidence for the conjecture. This program is joint work with Tom Mrowka.
“Nahm’s equations and transverse Hilbert schemes”
A construction of Atiyah and Hitchin produces hyperkähler metrics on open subsets of Hilbert schemes of points on a gravitational instanton (viewed as a complex surface). The resulting hyperkähler metrics are not the ones obtained from the Beauville-type construction on the full Hilbert scheme of points. Since the Atiyah-Hitchin construction is on the level of twistor spaces, the question of completeness of resulting metrics requires a different approach. I’ll show how to realise these “transverse Hilbert schemes” of points on several ALF spaces as moduli spaces of Nahm’s equations, which in particular proves completeness of the resulting metrics.
“Topology of the blow up set for the Seiberg–Witten monopoles with multiple spinors”
A sequence of the Seiberg-Witten monopoles with multiple spinors on a three-manifold can converge after a suitable rescaling to a Fueter section, say I, only on the complement of a subset Z. I will discuss the following question: Which pairs (I,Z) can (or can not) appear as the limit of a sequence of the Seiberg–Witten monopoles?