Georgia Institute of Technology College of Sciences

A map is a connected graph G embedded in a surface S (a closed 2-manifold) such that all components of S -- G are simply connected regions. A map is rooted if an edge is distinguished together with a direction on the edge and a side of the edge. Maps have been enumerated by both mathematicians and physicists as they appear naturally in the study of representation theory, algebraic geometry, and quantum gravity. In 1986 Bender and Canfield showed that the number of n-edge rooted maps on an orientable surface of genus g is asymptotic to t_g n^{5(g-1)/2}12n^n, (n approaches infinity), where t_g is a positive constant depending only on g. Later it was shown that many families of maps satisfy similar asymptotic formulas in which tg appear as \universal constants". In 1993 Bender et al. derived an asymptotic formula for the num- ber of rooted maps on an orientable surface of genus g with i faces and j vertices. The formula involves a constant tg(r) (which plays the same role as tg), where r is determined by j=i.In this talk, we will review how these asymptotic formulas are obtained using Tutte's recursive approach. Connections with random trees, representation theory, integrable systems, Painleve I, and matrix integrals will also be mentioned. In particular, we will talk aboutour recent results about a simple relation between tg(r) and tg, and asymptotic formulas for the numbers of labeled graphs (of various connectivity)of a given genus. Similar results for non-orientable surfaces will also be discussed.