# Richard Hain : Hodge theory and the Goldman-Turaev Lie bialgebra

In the 1980s, Bill Goldman used intersection theory to define
a Lie algebra structure on the free **Z** module *L(X)* generated by the
closed geodesics on a hyperbolic surface *X*. This bracket is related to
a formula for the Poisson bracket of functions on the variety of flat
*G*-bundles over *X*. In related work (1970s and 1990s), Vladimir Turaev
(with contributions by Kawazumi and Kuno in the 2000s) constructed a
cobracket on *L(X)* that depends on the choice of a framing. In this
talk, I will review the definition of the Goldman-Turaev Lie bialgebra
of a framed surface and discuss its relevance to questions in other
areas of mathematics. I'll discuss how Hodge theory can be applied to
these questions. I may also discuss some related questions, such as
the classification of mapping class group orbits of framings of a
punctured surface.

**Category**: Geometry and Topology**Duration**: 01:34:58**Date**: September 10, 2018 at 3:10 PM**Views**: 203-
**Tags:**seminar, Geometry/topology Seminar

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