Factorisation of permutations, and the Hurwitz Problem

David M. Jackson

University of Waterloo and MIT

February 18,
refreshments at 3:45pm


In this talk I shall discuss some tantalising and deep connexions between algebraic combinatorics and algebraic geometry, from my perspective as an algebraic combinatorialist.

An expression for the number of ramified covers of the sphere (elementary branch points, with the exception of one point (infinity) with arbitrary ramification) was conjectured by combinatorial means. It was discovered by using Hurwitz's encoding of a ramified cover as a transitive factorisation of a permutation into transpositions, with certain conditions. I shall discuss some of the algebraic combinatorics that lies behind the conjectured expression.

It is believed that the structure of the above problem, the classical Hurwitz Problem, is just a shadow of the rich structure that is thought to reside in the "double Hurwitz problem" (two points, zero and infinity, over which there is arbitrary ramification). This is a more complex permutation factorisation problem, but methods of algebraic combinatorics can be used to provide strong evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). I shall describe some of the combinatorial side of this research on double Hurwitz numbers.

This is joint work with Ian Goulden and Ravi Vakil.

Speaker's Contact Info: dmjackso(at-sign)math.mit.edu

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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

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