Speaker: Calvin McPhail-Snyder , Duke UniversityTitle: Hyperbolic tensor networks and the volume conjectureDate: 11/08/2022Time: 3:00 PM - 4:00 PMPlace: C304 Wells Hall (Virtual Meeting Link)Contact: Efstratia Kalfagianni (kalfagia@msu.edu)

Quantum invariants of links like the colored Jones polynomial (which arise from the quantum Chern-Simons theory of Witten-Reshetikhin-Turaev) have a purely algebraic construction in terms of the representation theory of quantum groups. Despite this algebraic nature they appear to be connected to geometry: a class of related volume conjectures assert that their semi-classical asymptotics determine geometric invariants like the hyperbolic volume. To better understand these conjectures a number of authors have studied ways to twist quantum invariants by geometric data. In particular, Blanchet, Geer, Patureau-Mirand, and Reshetikhin recently defined quantum holonomy invariants depending on a link in S^3 and a flat π°π©β connection on its complement. Their construction uses certain unusual cyclic modules of quantum π°π©β. For technical reasons the invariants are quite difficult to compute. In this talk (based on joint work with Nicolai Reshetikhin) I will explain how to effectively compute them using hyperbolic tensor networks constructed from quantum dilogarithms. Our construction reveals deep connections with hyperbolic geometry and suggests a way to break the Kashaev-Murakami-Murakami volume conjecture into two simpler pieces.

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