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.
Force and restraint
Simone Weil wrote about the Iliad, how it dealt so beautifully with notions of force.
Forty years on
It is forty years on from the Dismissal, or coup, that ended the Whitlam government.
Riddle. Mystery. Enigma.
I appear in a rather excellent and fun episode of the ABC Radio National program Radiotonic.
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.
Why your calculator is a weapon
I gave a talk about the Defence Trade Cooperation Act, encryption, and number theory, as part of the Monash University LunchMaths seminar series, in August 2015. Slides 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”.
Every world in a grain of sand: John Nash’s astonishing geometry
After the recent tragic death of John Forbes Nash Jr, many tributes have been paid to this great mathematician, who was made famous by the movie “A Beautiful Mind”, and much has been said about his work on game theory. But less has been said about Nash’s other mathematical achievements.
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”.