Topology, Combinatorics and Modular Forms (continued)

This year's lecture was a continuation of the previous one, which focused on the links between modular forms and algebraic geometry derived from the theory of Picard-Fuchs-type differential equations, but also discussed the links between arithmetic and topology as manifested through the quantum invariants of hyperbolic varieties in dimension 3 and knots. This second theme was one of the two topics covered this year. The other, closely related in its basic idea, but very different in terms of the objects studied and the methods employed, was the relationship between modular forms, q-hypergeometric functions and algebraic K-theory. The physicist Werner Nahm was responsible for the surprising discovery of this relationship, formulating a precise conjecture in a very specific context.