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Geometric recursion

Le : 19/12/2017 14h00
Par : Gaëtan Borot (MPIM, Bonn)
Lieu : I 001
Lien web :
Résumé : The topological recursion only rely on the structure of a combinatorial category related to surfaces by remembering the genus g and number of boundaries n and allowing abstract glueing between them, e.g. (g - 1,n +1) > (g,n) or (g_1,1 + n_1) * (g_2,1 + n_2) > (g_1 + g_2,n_1 + n_2). This is somehow frustrating as the geometry (present in most of the case of applications of the topological recursion) is lost. I will present a finer formalism, called geometric recursion, which builds inductively (from a small amount of initial data) functorial assignments from a category of surfaces with their mapping classes. This construction can take value in various topological spaces functorially attached to surfaces, e.g. functions, forms, sections of bundles of conformal blocks etc. over Teichmuller spaces, functions over the universal moduli space of flat connections, etc. If we choose functions from Teichmuller spaces of bordered surfaces valued in a Frobenius algebra, integrating the geometric recursion amplitudes over the moduli space, and taking the Laplace transform with respect to the boundary length, gives back the topological recursion. The prototype of such formulas are Mirzakhani-McShane identities and Mirzakhani recursion for the Weil-Petersson volumes of the moduli space. This is joint work with J.E. Andersen and N. Orantin.