Seminar on periods and motives | Is the GT Lie bialgebra motivic?

Abstract：

The Goldman--Turaev Lie bialgebra of an oriented 2-manifold $X$ is a Lie bialgebra structure on the free abelian group spanned by the conjugacy classes of its fundamental group. Its structure encodes how isotopy classes of immersed loops on $X$ intersect each other and themselves. When $X$ is a smooth complex curve, a suitable completion of the GT-Lie bialgebra carries a natural mixed Hodge structure. The bracket and cobracket are both morphisms (after a suitable twist). This raises the question of whether (when $X$ is defined over a number field) of whether the bracket and cobracket are motivic and, if so, how they are related to algebraic cycles.

In this talk I will define the Goldman bracket and Turaev cobracket, as well as their extensions by Kawazumi and Kuno. I'll survey what is known about the MHS and Galois actions on the completed GT-Lie bialgebra and indicate some connections to the Ceresa cycle when $g(X) > 2$.

About the speaker：

I am a topologist whose main interests include the study of the topology of complex algebraic varieties (i.e. spaces that are the set of common zeros of a finite number of complex polynomials). What fascinates me is the interaction between the topology, geometry and arithmetic of varieties defined over subfields of the complex numbers, particularly those defined over number fields. My main tools include differential forms, Hodge theory and Galois theory, in addition to the more traditional tools used by topologists.