Closed 3-manifolds and modularity

Contact: Christian Kaiser (kaiser@mpim-bonn.mpg.de)

In the early 90s, following some ideas of Witten, quantum invariants were successfully defined for 3-manifolds by Reshetikhin and Turaev. Witten's ideas from physics implied that these invariants should satisfy certain asymptotic properties. This leads to many conjectures such as the volume conjecture. For non-hyperbolic manifolds Lawrence and Zagier showed that the invariants are related to mock modular forms. More recently, the study of hyperbolic manifolds leads to quantum modular forms. I will describe a complete picture in a hyperbolic example, which relates the RTW invariant and new Zhat invariants predicted from physics.