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Title: Combinatorics of even-valent graphs on Riemann surfaces

Speaker: Roozbeh Gharakhloo

Affiliation: University of California, Santa Cruz

Abstract:
Let $\mathscr{N}_g(\mu,j)$ denote the number of connected labeled $\mu$-valent graphs with $j$ vertices that can be embedded in a compact Riemann surface of minimal genus~$g$. For example, $\mathscr{N}_0(4,1)=2$ and $\mathscr{N}_1(4,1)=1$, which can be verified by drawing connected labeled $4$-valent graphs with a single vertex on the sphere and on the torus. The problem of determining $\mathscr{N}_g(\mu,j)$ for fixed $g, j,$ and $\mu$ arose in models of two-dimensional quantum gravity and has since inspired a rich body of work, involving both combinatorial methods and techniques from random matrix theory.
   
After reviewing the connections with random matrices, orthogonal polynomials and Riemann-Hilbert problems, I will present existing results for $\mathscr{N}_g(\mu,j)$ as a function of $j$ when $g$ and $\mu$ are both fixed. I will then describe recent progress on obtaining explicit formulae for $\mathscr{N}_g(2\nu,j)$, viewed as a function of both $j$ and $\nu$ for fixed $g$. If time permits, I will also discuss new developments on the combinatorics of mixed-valent graphs.
   
This talk is based on joint works with A. Barhoumi (Grinnell), P. Bleher (IU), N. Hayford (KTH), T. Lasic Latimer (UCSC), and K. McLaughlin (Tulane).