On 11 December, 2014 I gave a talk at the 8th Australia New Zealand Mathematics Convention, at the University of Melbourne, as part of the Geometry and Topology session. Slides are available.
Strings, fermions, curves on surfaces, Unimelb Oct 2014
On 24 October, 2014 I gave a talk at the University of Melbourne, for the Algebra-Geometry-Topology Seminar. Slides are available.
Strings, fermions and the topology of curves on annuli
In previous work with Schoenfeld, we considered a string-type chain complex of curves on surfaces, with differential given by resolving crossings, and computed the homology of this complex for discs.
In this paper we consider the corresponding “string homology” of annuli. We find this homology has a rich algebraic structure which can be described, in various senses, as fermionic. While for discs we found an isomorphism between string homology and the sutured Floer homology of a related 3-manifold, in the case of annuli we find the relationship is more complex, with string homology containing further higher-order structure.
Topological recursion and a quantum curve for monotone Hurwitz numbers
Joint with Norman Do and Alastair Dyer.
Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In this paper, we prove that monotone Hurwitz numbers arise from the topological recursion of Eynard and Orantin applied to a particular spectral curve. We furthermore derive a quantum curve for monotone Hurwitz numbers. These results extend the collection of enumerative problems known to be governed by the paradigm of topological recursion and quantum curves, as well as the list of analogues between monotone Hurwitz numbers and their classical counterparts.
Discrete contact geometry, May 2014
On 12 May, 2014 I gave a talk at the Monash University Discrete Mathematics Seminar. Slides are available.
An explicit formula for the A-polynomial of twist knots
We extend Hoste-Shanahan’s calculations for the A-polynomial of twist knots, to give an explicit formula.
Twisty itsy bitsy topological field theory
We extend the topological field theory (“itsy bitsy topological field theory”‘) of our previous work from mod-2 to twisted coefficients. This topological field theory is derived from sutured Floer homology but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has information-theoretic (“itsy”) and quantum-field-theoretic (“bitsy”) aspects. In the process we extend some results of sutured Floer homology, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.
A Yang-Baxter equation from sutured Floer homology, Sep 2013
On 30 September, 2013 I gave a talk at the 57th Australian Mathematical Society Annual Meeting, at the University of Sydney, as part of the Geometry and Topology session. Slides are available.
Sutures, quantum groups, TQFT, May 2013
On 24 May, 2013 I gave a talk at the University of Melbourne, for the Algebra-Geometry-Topology Seminar. The talk was entitled “Sutures, quantum groups and topological quantum field theory”. Slides are available.
Contact topology and elementary combinatorics, April 2013
On 16 April, 2013 I gave a talk at ANU, for the Algebra and Topology seminar. The talk was entitled “Contact topology and holomorphic invariants via elementary combinatorics”. Slides are available.