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)

Titel: Introductory lectures on exotic spheres

One of the milestones in differential topology is Milnor’s discovery of exotic spheres (that is, smooth manifolds that are homeomorphic but not diffeomorphic to the standard sphere) and his subsequent classification of them in homotopy-theoretic terms together with Kervaire.

Assuming only basic knowledge in algebraic and differential topology, this minicourse aims to explain these results and their proofs in detail. This will involve touching on a variety of key techniques in manifold topology along the way, such as surgery, characteristic classes, bordism theory, and stable homotopy theory.