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Abstract |
The talk will focus on quantum modularity relations satisfied by the q-Pochhammer symbol $(q)_N = (1-q) ... (1-q^N)$ at $q=\exp(2 \pi i x)$. These formulas can be interpreted as finite analogues of the usual modularity for the Dedekind eta-function. We'll discuss certain aspects which come very handy upon summing over $N$. We'll explain how these can be used, in the context of Kashaev's invariant of hyperbolic knots, to prove, in a few cases, Zagier's quantum modularity conjecture by means of what we currently know on the Volume Conjecture. This is based on joint work with Sandro Bettin. |
