## EVENT DETAILS AND ABSTRACT

**Geometry/Physics Seminar**
**Title:** Quantum invariants of surface diffeomorphisms and 3-dimensional hyperbolic geometry

**Speaker:** Francis Bonahon

**Speaker Info:** USC

**Brief Description:**

**Special Note**:

**Abstract:**

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.

**Date:** Thursday, November 04, 2021

**Time:** 4:00pm

**Where:** Lunt 107

**Contact Person:**

**Contact email:** hyuan@northwestern.edu

**Contact Phone:**

Copyright © 1997-2024
Department of Mathematics, Northwestern University.