On the Hurwitz Enumeration Problem
and some related combinatorial and geometric questions
University of Waterloo
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
Speaker's Contact Info: dmjackson(at-sign)math.uwaterloo.ca
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