Mock modular and quantum modular forms

While the theory of mock modular forms has seen great advances in the last
decade, questions remain. In this talk, we revisit Ramanujan’s last letter
to Hardy, and prove one of his remaining conjectures as a special case of a
more general result. We will show how quantum modular forms, defined by
Zagier in 2010, underlie new relationships between well known combinatorial
modular and mock modular forms. We also show how Rogers-Fine false theta
functions, functions that have not been well understood within the theory
of modular forms, specialize to quantum modular forms. This is joint work
with Ken Ono (Emory U.) and Rob Rhoades (CCR-Princeton).