# Basic Notions: Ruth Lawrence (HUJI) The Quantum Modularity Conjecture

Date:
Thu, 14/10/202116:00-17:15
Location:
Ross 70
Over the last 20 years, many examples have arisen, mainly from quantum invariants of knots and 3-manifolds, of "expressions" in a variable q which are well-defined both at all roots of unity and as formal (divergent) power series or asymptotic expansions around q=1 (or more generally any root of unity). The simplest example of such an expression is $\sum_{n=0}^\infty (1-q)\ldots(1-q^n)$ (studied in Zagier [Topology 40 (2001) 945-960]) which is on the one hand related to the combinatorics of Vassiliev invariants and on the other to the Dedekind eta-function. Another early example (Lawrence & Zagier [Asian J. Math 3 (1999) 93-108]) is the values of the quantum sl_2 invariant (the Witten-Reshekhin-Turaev invariant) of the Poincare homology sphere, which is defined at roots of unity, and by its connection with Witten-Chern-Simons theory has a perturbative expansion also around q=1. In both cases, connections were shown with "almost modular forms", meaning that these are not modular forms but the error from being modular is given by something controllable.