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

Thu, 14/10/202116:00-17:15
Ross 70
link to video:


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.
In an attempt to produce a theory at once formalising these properties and able to encompass more general examples, Zagier introduced the notion of Quantum Modular Forms [Clay Mathematics Proceedings 12 (2010)].

We will discuss work of Garoufalidis & Zagier (involving also work of Neumann, Gukov, Dimofte, Kashaev and many others) on the Quantum Modularity Conjecture and various generalizations and refinements, which conjecturally exhibit (based on an enormous amount of numerical evidence in particular cases) a remarkable fine structure in the Kashaev invariant of hyperbolic knots, including holomorphic, modular and asymptotic structure as well as the unexpected appearance of certain number fields and special units in them.
This is related to Nahm's conjecture, proved by Calegari, Garoufalidis & Zagier [],
which gives a necessary condition for the modularity of certain q-hypergeometric series in terms of the vanishing of a constructed element of the Bloch group.

The lecture will be self-contained and we will not assume any prior knowledge of quantum invariants - in fact we will hardly mention topology, use a little geometry of triangulations and will mainly be concerned with the algebraic q-series obtained and their properties.