Seminar: The Symplectic Structure of Representation Varieties

Wintersemester 2018/2019

Prof. Anna Wienhard, Prof. Peter Albers, Mareike Pfeil

Announcement

Please register on Muesli in order to obtain mails concerning the seminar.

Time and Location

Thursdays, 9-11am, SR 4

Next Talk - Irregular time and place!

Tuesday, 05.02.19, 8.15 - 9.25 am, Conference room, 5th floor

Description

In 1984 William Goldman showed that the variety of representations of the fundamental group of a topological surface into any reductive Lie group carries a natural symplectic structure on its smooth part.
In this seminar we want to discuss first the construction of this symplectic structure both from the perspective in Goldman's paper as well as via symplectic reduction as described by Jeffrey and Weitsman. Then we will further explore the symplectic geometry and symplectic dynamics of representation varieties.

Schedule

Date	Speaker	Topic
18.10.2018	-	Organizational meeting
24.10.2018	Arnaud	Goldman [1] - The definition of the 2-form
01.11.2018	-	Public Holiday
08.11.2018	Lukas	Symplectic reduction I
15.11.2018	Lukas	Symplectic reduction I - continued
22.11.2018	Arnaud	Goldman [1] - closedness
29.11.2018	Arnaud	Goldman [1] - continued
06.12.2018	Johannes	Symplectic reduction II
13.12.2018	Milan	Delzant's Garden - A classification of symplectic toric manifolds
20.12.2018	Milan	Delzant's Garden - continued
10.01.2019	Andy	Poisson geometry of relative character varieties and compact components for SL(2,R) [5]
17.01.2019	Andy	Poisson geometry of relative character varieties and compact components for SL(2,R) - continued [5]
24.01.2019	Mareike	Goldman [2] - invariant functions
31.01.2019	Mareike	Goldman [2] - invariant functions - continued
05.02.2019	Mareike	Goldman [2] - Hamiltonian flows

Literature

[1] William M. Goldman The symplectic nature of fundamental groups of surfaces. Advances in Mathematics, 52(2):200-225, 1984.
[2] William M. Goldman. Invariant functions on Lie groups and hamiltonian flows of surface group representations. Inventionen mathematicae, 85(2):263-302, 1986.
[3] Lisa C Jeffrey. Flat connections on oriented 2-manifolds. Bulletin of the London Mathematical Society, 37(1):1-14, 2005.
[4] Lisa C Jeffrey and Jonathan Weitsman. Toric structures on the moduli space of flat connections on a Riemann surface: volumes and the moment map. Advances in mathematics , 106(2):151-168, 1994.
[5] Bertrand Deroin and Nicolas Tholozan Super-maximal representations from fundamental groups of punctured surfaces to PSL(2,R) arXiv:1604:00330

Last update: 04.02.2019