Fullerenes and modular forms

Fullerenes are polyhedral molecules made of carbon atoms, first discovered in 1985. There is a simple mathematical model of them, as a convex polyhedron all of whose faces are hexagons and pentagons. During their wave of popularity in the 1990s, mathematical chemists developed several (slow) algorithms to enumerate them, and asked how many isotopes exist with a given number of carbon atoms. In joint work with Phil Engel, we show that these numbers are the Fourier coefficients of an Eisenstein series, and find an explicit formula. In this talk I will try to tell some of the chemistry story, as well as explaining the math.