Maps and the Jack symmetric functionDavid JacksonUniversity of Waterloo
February 20,

ABSTRACT


A map is a 2cell embedding of a graph in a surface. The determination of the counting series for maps is a question that arises in combinatorics, in geometry, and in physics as a partition function. Tutte, for example. studied the enumeration of maps in the sphere as part of his attack on the Four Colour Problem. In this talk I will discuss an algebraic approach to determining the counting series for maps in orientable surfaces and in all surfaces (including nonorientable surfaces) in terms of Jack symmetric functions. The indeterminate b in the Jack parameter a=1+b is conjectured to mark an invariant of rooted maps. The counting series for maps with one vertex is central to the determination of the Euler characteristic of the moduli spaces of complex (b=0; Schur functions) and real algebraic curves (b=1; zonal polynomials). The complex case was treated by Harer and Zagier. If time permits I shall refer briefly to the conjectured combinatorial role of b in this context. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

