$N=4$ dyons and Mock modular forms

We show that the generating function for the quantum degeneracies of black holes in N = 4 string theories in four dimensions is a mock modular form. In this talk, I will explain the mathematics behind this statement, as well as its implications for the physics of supersymmetric black holes. I will then discuss the connection with wall crossing and holography on the physics side. I will end by presenting some identities involving the mock modular forms arising from the black hole problem and older known mock modular forms. This is joint ongoing work with A. A. Dabholkar and D.Zagier.