Aula Riunioni - Dipartimento di Matematica
Quantum invariants of three manifolds were discovered
and defined in the early 1990s by Witten-Reshetikhin-Turaev. The
physical approach to these invariants suggested they satisfy interesting
links to the geometry of three manifolds through perturbation theory.
On the other hand the mathematical description does not make these
links at all clear. In exploring these perturbative properties numerically,
Zagier noticed that these quantum invariants satisfy strange and new kinds
of modular properties. This has led to the idea of quantum modular forms
understood in subsequent work of Garouflaidis-Zagier. Then work of
Garoufalidis-Gu-Mariño has used this idea of quantum modularity to give a
method of computing Borel resummation of associated asymptotic series
and their Stokes phenomenon. This story has been studied in the case of
some simple hyperbolic knots and I will describe this, along with the extension
to the case of simple closed hyperbolic three manifolds.