組合研討會

主講者: Prof. Jason Zhicheng Gao (Carleton University)
講題: Surface Maps, Matrix Integral, KP-hierarchy, Painleve I, and Quantum Gravity
時間: 2008-12-30 (Tue.)  10:30 - 11:30
地點:
Abstract: Let $\sum_g$ be the orientable surface of genus $g$. A $map$ on $\sum_g$ is a graph $G$ embedded on $\sum_g$ such that all components of $\sum_g-G$ are simply connected regions. These components are called $faces$ of the map. A map is rooted by distinguishing an edge, an end vertex of the edge and a side of the edge. Surface maps have been studied extensively in combinatorics and they also appear naturally in quantum physics in the context of matrix integrals. Let $M_{n,g}({\cal F})$ be the number of rooted maps with $n$ edges on $\Sigma_g$ in a family $\cal F$. Many families of maps satisfy the following pattern
$M_{n,g}({\cal F})\sim \alpha t_g(\beta n)^{5(g-1)/2}\gamma^n,~n\to \infty,$
where $\alpha,\beta$ and $\gamma$ are constants depending only on $\cal F$, and $t_g$ depends only on $g$. For over twenty years, the constants $t_g$ remained hard to compute or estimate partly due to the complexity of their nonlinear recursion. In this talk, we review some historical results in enumerative map theory and describe some connections among surface maps, matrix integral, KP-hierarchy of quadratic PDEs, Painleve I ODE, leading to an asymptotic formula for $t_g$. This asymptotic formula for $t_g$ plays an important role in the recent proof of a conjecture of $^{\prime}t$ Hooft about analyticity of free energy.
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