Ramanujan, the Bloch group, and modularity

Frank Calegari (University of Chicago)
Fri 24 Feb 2017 at 1pm in room 213 (Peter Hall building)

The Rogers-Ramanujan identity $$1 + \frac{q}{(1-q)} + \frac{q^4}{(1-q)(1-q^2)} + \frac{q^9}{(1-q)(1-q^2)(1-q^3)} + \dots = \frac{1}{(1-q)(1-q^4)(1-q^6)(1 - q^9)\dots}$$ says that a certain $q$-hypergeometric function (the left hand side) is equal to a modular function (the right hand side, up to a power of $q$). There are a number of generalizations of this identity coming from Conformal Field Theory. To what extent can one classify all $q$-hypergeometric functions which are modular? We discuss this question and its relation to conjectures in knot theory and K-theory. This is joint work with Stavros Garoufalidis and Don Zagier.