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Friday, September 26, 2014

**Abstract:** The talk will be a survey of rigorous results on the all-order asymptotic expansion in matrix models : the existence of a 1/N expansion or oscillatory expansions (established by probabilistic methods), and the topological recursion that govern the combinatorial/algebro-geometric structure of the expansion. I shall describe two applications : (1) the derivation of all-order asymptotic expansions for solutions of the Toda chain and (skew) orthogonal polynomials ; (2) some analyticity results for perturbative knot invariants in Seifert manifolds. This is based on joint works with Guionnet, Kozlowski, and Eynard and Orantin.