In October-November 2016 I gave two talks at the MSI Workshop on Low-Dimensional Topology & Quantum Algebra at ANU, Canberra. Some slides are available.
Strand algebras and contact categories
We demonstrate an isomorphism between the homology of the strand algebra of bordered Floer homology, and the category algebra of the contact category introduced by Honda. This isomorphism provides a direct correspondence between various notions of Floer homology and arc diagrams, on the one hand, and contact geometry and topology on the other. In particular, arc diagrams correspond to quadrangulated surfaces, idempotents correspond to certain basic dividing sets, strand diagrams correspond to contact structures, and multiplication of strand diagrams corresponds to stacking of contact structures. The contact structures considered are cubulated, and the cubes are shown to behave equivalently to local fragments of strand diagrams.
Talk on hyperbolic volume and Mahler measure, April 2016
On 8 April, 2016 I gave a talk at the University of Melbourne in the Knot Invariants seminar. The talk was entitled “Hyperbolic volume and the Mahler measure of the A-polynomial”
Talk on trinities, hypergraphs, contact structures, March 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.
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.
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.