Projects per year
Abstract
We consider an elementary, and largely unexplored, combinatorial problem in lowdimensional topology: for a compact surface S, with a finite set of points F fixed on its boundary, 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. We show that such curve counts obey an effective recursion, in the general spirit of topological recursion, and exhibit quasipolynomial behavior. This "elementary curvecounting" is in fact related to a more advanced notion of "curvecounting" 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 quasipolynomials governing the enumerative problem encode intersection numbers on moduli spaces. 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 and quantum curves.
Original language  English 

Article number  1750012 
Number of pages  105 
Journal  International Journal of Mathematics 
Volume  28 
Issue number  2 
DOIs  
Publication status  Published  1 Feb 2017 
Keywords
 Catalan numbers
 Curves on surfaces
 Topological recursion
Projects
 2 Finished

Quantum invariants and hyperbolic manifolds in threedimensional topology
Australian Research Council (ARC), Monash University
1/01/16 → 31/07/20
Project: Research

The geometry and combinatorics of moluli spaces
Australian Research Council (ARC)
30/06/13 → 30/08/18
Project: Research