The University of Michigan Combinatorics Seminar
The b-conjectures concern the conjectured existence of a combinatorial invariant associated with graphs in orientable and non-orientable surfaces (sphere, torus,..., projective plane, Klein bottle,...), drawn so that edges meet only at vertices. Such configurations are called maps. On deletion of the edges, the surface decomposes into discs, the faces of the map. Maps occur in various areas of mathematics and mathematical physics (as Feynman diagrams, for example).
I will outline how the b-conjectures arise and how the ideas are related to the Hurwitz enumeration problem (1895) concerning the number of ways of factorising a permutation in S_n into a product of transpositions that generate S_n. The parameter of the Jack symmetric function plays a crucial role in the development of these conjectures, and is closely related to the "b" in the title. (The factorisations encode certain ramified coverings of the sphere.)
Recently there have been significant extensions of the Hurwitz problem, relating it to Witten's Conjecture, and suggesting that aspects of the latter, at least in principle, may be looked at from a combinatorial perspective.
The talk is intended to give both a bird's eye view of this area of combinatorics, and a number of explicit results. Details of the proofs of the latter are available in material that will be cited (or at http://www.math.uwaterloo.ca/~dmjackso as .ps files).
This talk includes joint work with Gilberto Bini, Ian Goulden, John Harer, Ravi Vakil, and T.Visentin.