Quantum modularity and closed three-manifolds (Doctoral defense)

Over the last two decades, new forms of modularity were observed in quantum invariants of knots and three manifolds. The first truly new example came from the invariants of the simplest hyperbolic knot studied by Zagier. Recently, Garoufalidis and Zagier gave an extensive study of three hyperbolic knots giving a clear picture of this new kind of modularity. I will describe how completely analogous statements hold for the invariants of a closed three manifold and how this unifies various conjectures on the behaviour of these invariants. The main technical tools come from work of Garoufalidis and Kashaev on integrals of the quantum dilogarithm.

For zoom details contact: Campbell Wheeler (cjwh @ mpim-bonn.mpg.de)