Lecture 1, Lecture 2 (April, 23th | May, 28th | June, 25th)

Alexander Thomas (Heidelberg)

Title: The geometry of character varieties

Abstract: Character varieties describe the representation theory of the fundamental group of a manifold M. They play an important role in Geometry, Algebra and Theoretical Physics. In these lectures, we will discuss various geometric aspects, each of which corresponds to one of the 6 lectures. The emphasis is on ideas, motivations and links to divers areas of mathematics, without entering technical details.
1) Algebraic Geometry: The algebraic geometric viewpoint leads to the geometric invariant theory (GIT) and allows to properly define the character varieties.
2) Differential Geometry: The Riemann-Hilbert correspondence allows to identify character varieties with the space of flat connections modulo gauge transformations.
3) Symplectic Geometry: When M is a surface, the Atiyah-Bott reduction gives a natural symplectic structure on character varieties.
4) Hyperbolic Geometry: When M is a surface, the moduli space of hyperbolic structures, the Teichmüller space, is a connected component of some character variety.
5) Complex Geometry: When M is a Kähler manifold, the non-abelian Hodge theory allows to describe character varieties via holomorphic objects, called Higgs bundles.
6) Hyperkähler Geometry: The proof of the nonabelian Hodge correspondence is best understood in the context of twistor spaces of hyperkähler manifolds.

Lecture 1, Lecture 2 (May, 7th | June, 11th | July, 9th)

Manuel Krannich (Karlsruhe)