Marco Bertola

Symplectic structure on the SL_2(R) character variety of closed Riemann surfaces and Darboux coordinates.

The $SL_n$ representation space of the fundamental group of a Riemann surface admits a Poisson (or symplectic) structure due to W. Goldman. When specialized to $SL_2(R)$ this symplectic structure coincides with the Weil-Petersson form. A famous result of Scott Wolpert proved that the Fenchel-Nielsen coordinates are Darboux coordinates for this symplectic form.

In this talk I will explain what the character variety is in terms of explicit matrix representations, and how to provide alternative ``log-canonical’’ coordinates (i.e. coordinates in terms of the logarithms of which the symplectic form has constant coefficients).
These coordinates are provided in terms of triangulations in the work of Fock-Chekov-Thurston, but only in the case of surfaces with at least one boundary.
I will then explain how to use a process of surgery to obtain similar coordinates on the Character variety of closed surfaces. This is a joint work with D. Korotkin and J. Pillet.

Royal Society Wolfson visiting professor