Description of the course:
I will present joint work with Frank Calegari and Stavros Garoufalidis in which elements of the Bloch group of a number field (a more elementary version of algebraic K-theory that will be explained in detail in the lectures) produce units in cyclotomic extensions of this field. The result was motivated by experimental discoveries relating to quantum invariants (specifically, Jones polynomials and the Kashaev invariant) of knots, and had an unexpected application to a proof of a well-known conjecture of Werner Nahm relating the modularity of certain q-hypergeometric series to the vanishing of a certain invariant in the Bloch group.