Mathematics Colloquium

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract setup we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies and Zeta-functions based on the simpel closed geodesic length spectrum. We shall see how Geometric Recursion provides us with a kind of categorification of Topological Recursion, namely any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. We will end the talk by applying the machinery to obtain interesting results on expectation values of various statistics of length of simple closed geodesic over moduli spaces of hyperbolic surfaces. The work presented is joint with G. Borot and N. Orantin.