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Weil–Petersson volumes and cone surfaces
The moduli spaces of hyperbolic surfaces of genus g with n geodesic boundary components are naturally symplectic manifolds. Mirzakhani proved that their volumes are polynomials in the lengths of the…
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Quantum curves for the enumeration of ribbon graphs and hypermaps
The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite…
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Monotone Orbifold Hurwitz Numbers
In general, the Hurwitz numbers count the branched covers of the Riemann sphere with prescribed ramification data or, equivalently, the factorizations of a permutation with prescribed cycle structure…
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Topological recursion and a quantum curve for monotone Hurwitz numbers
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Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes
Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. Mirzakhani proved that Weil-Petersson volumes exhibit polynomial behaviour and that their…
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Bounds on the max and min bisection of random cubic and random 4-regular graphs
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Intersection theory on moduli spaces of curves via hyperbolic geometry
This thesis explores the intersection theory onMg,n, the moduli space of genus g stable curves with n marked points. Our approach will be via hyperbolic geometry and our starting point will be the…
Bisection of Random Cubic Graphs
TLDR
Two randomized algorithms are presented to bound the bisection width of random n-vertex cubic graphs and an asymptotic upper bound and a corresponding lower bound are obtained.
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The asymptotic Weil-Petersson form and intersection theory on M_{g,n}
Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths…
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Orbifold Hurwitz numbers and Eynard-Orantin invariants
We prove that a generalisation of simple Hurwitz numbers due to Johnson, Pandharipande and Tseng satisfy the topological recursion of Eynard and Orantin. This generalises the Bouchard-Marino…
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