Department of Mathematics

Geometry and Topology

  •  Francis Bonahon, USC
  •  Quantum invariants of surface diffeomorphisms and 3-dimensional hyperbolic geometry
  •  10/19/2021
  •  3:00 PM - 4:00 PM
  •  Online (virtual meeting) (Virtual Meeting Link)
  •  Honghao Gao (gaohongh@msu.edu)

This talk is motivated by surprising connections between two very different approaches to 3-dimensional topology, namely quantum topology and hyperbolic geometry. The Kashaev-Murakami-Murakami 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 n-th Jones polynomial of the knot at the n-root 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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Michigan State University
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