Topology, Combinatorics and Modular Forms

The aim of this year's lecture (and its planned extension for next year) was, as the name suggests, to make the links between rather heterogeneous mathematical disciplines involving Combinatorics properties of some kind. These links often appear in relation to quantum physics. The two main axes are, on the one hand, the link between modular forms and algebraic geometry stemming from mirror symmetry, manifested through differential equations of the Picard-Fuchs type, and, on the other hand, the link between arithmetic and topology, manifested through quantum invariants of hyperbolic varieties in dimension 3 and knots.