Date: Thursday, November 11, 2021
Location: 3866 East Hall (4:00 PM to 5:00 PM)
Title: Quantum invariants of surface diffeomorphisms and 3dimensional hyperbolic geometry
Abstract: This talk is motivated by surprising connections between two very different approaches to 3dimensional topology, namely quantum topology and hyperbolic geometry. The KashaevMurakamiMurakami Volume Conjecture connects the growth of colored Jones polynomials of a knot to the hyperbolic volume of its complement. More precisely, for each integer n, one evaluates the nth Jones polynomial of the knot at the nroot of unity exp(2 pi i/n). The Volume Conjecture predicts that this sequence grows exponentially as n tends to infinity, with exponential growth rate related to the hyperbolic volume of the knot complement.
I will discuss a closely related conjecture for diffeomorphisms of surfaces, based on the representation theory of the Kauffman bracket skein algebra of the surface, a quantum topology object closely related to the Jones polynomial of a knot. I will describe the mathematics underlying this conjecture, which involves a certain Frobenius principle in quantum algebra. I will also present experimental evidence for the conjecture, and describe partial results obtained in work in progress with Helen Wong and Tian Yang.
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Speaker: Francis Bonahon
Institution: University of Southern California
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