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Poisson Geometry Program Seminar: Mini course by Moira Chas

Time: Jun 14, 2018 10:00 AM

Speaker: Dennis Sullivan

Title: The generalization of the Goldman bracket to Three dimensions and its relation to Geometrization

This lecture begins with some background to the joint work of Moira Chas and Siddartha Gadgil recognizing “geometrization”. We describe elementary pictorial computations of the zeroth, first and second homology of the space of unparametrized generic closed curves in a compact three manifold.The degree one piece has a pictorial lie algebra structure which acts as a lie algebra on each piece.This structure is part of “String Topology”. arXiv:math/9911159v1 . See also wikipedia String Topology.

The second part of the lecture describes how this structure in degrees zero and one plus the power operations in degree zero recognizes key features of the Geometrization, the above mentioned joint work. The lions share of effort concerns the torus decomposition of three manifolds which carry mixed geometry. One application would tell whether or not a link or knot complement in the three sphere has a positive hyperbolic volume. An interesting research possibility is to try to use this relationship to attack the volume conjecture of Kashaev and Murikami. This conjecture relates the asymptotics of quantum topology knot invariants at roots of unity to the hyperbolic nature of the knot or link complement. The conjecture is based on computer computations.See wikipedia Volume Conjecture.