Black holes, shadows, and moonshine
- Don Zagier (Max-Planck-Institut für Mathematik, Bonn, Germany)
In his last letter to Hardy, four months before his early death in 1920, Ramanujan gave a list of 17 power series that he called "mock theta functions" and that he was sure would eventually become important in mathematics. An understanding of the properties of these functions and their generalizations ("mock modular forms") came only in 2002 with the thesis of Sander Zwegers, who showed that they have a weakened modular transformation property with an obstruction to true modularity that is given by an auxiliary function called the "shadow" and which is itself a modular form.
More recently it has transpired that these mock modular forms also appear naturally in physics, e.g. in the string theory of black holes. Even more recently they have also occurred in the discovery of new varieties of "Moonshine" (Mathieu moonshine, umbral moonshine,...) generalizing the famous Monstrous Moonshine of the 80s. We will give a survey of some of these developments.