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I. P. Goulden

Publications and source records attributed to I. P. Goulden.

22 records · Page 2Linked to original sources
arXiv1999

The number of ramified coverings of the sphere by the double torus, and a general form for higher genera

An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to prove a conjecture of Graber and Pandharipande, giving a linear recurrence equation for the number of these coverings with no ramification over infinity. The general form of the series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two.

math.AG↗
arXiv1999

Transitive factorisations in the symmetric group, and combinatorial aspects of singularity theory

We consider the determination of the number $c_k(α)$ of ordered factorisations of an arbitrary permutation on n symbols, with cycle distribution $α$, into k-cycles such that the factorisations have minimal length and such that the group generated by the factors acts transitively on the n symbols. The case k=2 corresponds to the celebrated result of Hurwitz on the number of topologically distinct holomorphic functions on the 2-sphere that preserve a given number of elementary branch point singularities. In this case the monodromy group is the alternating group, and this is another case that, in principle, is of considerable interest. We conjecture an explicit form, for arbitrary k, for the generating series for $c_k(α)$, and prove that it holds for factorisations of permutations with one, two and three cycles ($α$ is a partition with at most three parts). The generating series is naturally expressed in terms of the symmetric functions dual to those introduced by Macdonald for the ``top'' connection coefficients in the class algebra of the symmetric group. Our approach is to determine a differential equation for the generating series from a combinatorial analysis of the creation and annihilation of cycles in products under the minimality condition.

math.CO↗
arXiv1999

A geometric parametrization for the virtual Euler characteristic for the moduli spaces of real and complex algebriac curves

We show that the virtual Euler characteristics of the moduli spaces of $s$-pointed algebraic curves of genus $g$ can be determined from a polynomial in $1/γ$ where $γ$ permits specialization, through $γ=1,$ to the complex case treated by Harer and Zagier and, through $γ=1/2$, to the real case. This polynomial appears to have geometric significance, and may be the virtual Euler characteristic of some moduli space, as yet unidentified. This is related to a conjecture that the indeterminate $b=γ^1-1$ is associated with a combinatorial invariant of cell-decompositions through matrix models and the Jack symmetric functions. The development uses Strebel differentials to triangulate the moduli spaces, and the identification of $γ$ both as a parameter in a Jack symmetric function and as a parameter in a matrix model through generalized Selberg integrals.

math.AG↗