On the Hurwitz Enumeration Problem and some related combinatorial and geometric questions

David Jackson

University of Waterloo

March 17,
refreshments at 3:45pm


The problem of finding explicitly the number of ramified coverings of the sphere by a surface of genus g, with simple branch points, given degree and prescribed ramification over infinity is a classical problem in geometry. It is often referred to as the "Hurwitz Enumeration Problem." From a combinatorial point of view, the problem is equivalent to counting the number of certain types of ordered transitive factorizations of a permutation into transpositions, a fact observed by Hurwitz in his 1895 paper. Factorizations of permutations into transpositions also occur in the counting of embeddings of graphs, and therefore in connection with the moduli spaces of real and complex algebraic curves. In this talk I will describe a combinatorial approach to the problem that used a "cut-and-join" analysis, the treatment of the formal partial differential equation that is obtained, and the way in which results conjectured by these means can be proved. I will also discuss a number of questions about certain moduli spaces that this combinatorial approach raises. This material involves pieces of joint work, at different times, with Ian Goulden, John Harer, and Alek Vainstein. Preprints can be obtained from http://xxx.lanl.gov/ps/math.AG/9902009, AG/9902011 and AG/9902044. A related talk will also be given on March 1 in the Applied Math Colloquium which may serve as a more elementary introduction.

Speaker's Contact Info: dmjackson(at-sign)math.uwaterloo.ca

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

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