Theory and applications of false modular forms

In his famous last letter to Hardy, sent just three months before his untimely death in 1920, Ramanujan informed him of his discovery of a new class of functions which he called "  mockϑ-functions " and of which he was convinced that "they enter into mathematics as beautifully as the ordinary ϑ-functions" - i.e., in modern terminology, as the classical modular forms. For more than 80 years these objects remained mysterious, but in 2002 the decisive property of the functions found by Ramanujan was detected by Sander Zwegers, and since then there has been a veritable explosion of activity, with theoretical developments of the subject and numerous applications in various fields of mathematics and theoretical physics, and we can now say that Ramanujan's prediction has proved correct. Mock modular forms, to give them their current name, were already treated peripherally in the 2005-2006 lecture, and were also dealt with in the part of the 2011-2012 lecture that took place in South Korea. In this year's lecture, the theory was presented in much greater detail, with the emphasis on links with Jacobi's theory of modular forms and physics-motivated applications (string theory of black holes). All the results presented are contained in a 150 page paper (arXiv:1208.4074) written with Atish Dabholkar and Sameer Murthy, which will appear as a monograph.