On 14 March, 2016 I gave a talk at Monash University in the Discrete Mathematics seminar. The talk was entitled “Trinities, hypergraphs, and contact structures”. Slides from the talk are available.
Trinities, SFH, contact structures, Kioloa Jan 2016
On 14 March, 2016 I gave a talk at Monash University in the Discrete Mathematics seminar. The talk was entitled “Trinities, hypergraphs, and contact structures”. Slides from the talk are available.
Counting curves on surfaces
Joint with Norman Do and Musashi Koyama.
In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface S, and fix a number of points F on its boundary. We ask: how many configurations of disjoint arcs are there on S whose boundary is F?
We find that this enumerative problem, counting curves on surfaces, has a rich structure. For instance, we show that the curve counts obey an effective recursion, in the general framework of topological recursion. Moreover, they exhibit quasi-polynomial behaviour.
This “elementary curve-counting” is in fact related to a more advanced notion of “curve-counting” from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative problem encode intersection numbers on moduli spaces. Furthermore, among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions, topological recursion, and quantum curves.
Counting curves on surfaces, AustMS Sep 2015
On 30 September, 2015 I gave a talk at the Australian Mathematical Society Annual Meeting at Flinders University, in Adelaide. The talk was entitled “Counting curves on surfaces”. Slides from the talk are available.
Introduction to the AJ conjecture, Melbourne, August 2015
On 14 August 2015, I gave a talk in the Melbourne University Knot Invariant seminar, entitled “Introduction to the AJ conjecture”.
Geometric quantisation and A-polynomials, June 2015
On 12 June, 2015 I gave a talk at the University of Melbourne, in the Moduli Spaces seminar. The talk was entitled “Geometric quantisation and calculation of A-polynomials”.
The A-polynomial, symplectic geometry, and quantisation, May 2015
On 15 May, 2015 I gave a talk at the University of Melbourne, in the Moduli Spaces seminar. The talk was entitled “The A-polynomial, symplectic geometry, and quantisation”.
Contact topology and holomorphic invariants, Tokyo Feb 2015
On 18 February, 2015 I gave a talk at Tokyo Institute of Technology. The talk was entitled “Contact topology and holomorphic invariants via elementary combinatorics”. Slides are available.
Strings, fermions, curves on surfaces, ANZMC Dec 2014
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.