Mini Course on Character Varieties and their applications in Quantum Topology

Mini Course on Character Varieties and their applications in Quantum Topology

by Adam Sikora

Character varieties are moduli spaces of representations of discrete groups into algebraic ones. In "classical" geometry and low dimensional topology they appear as the moduli spaces of geometric structures, moduli spaces of principal bundles, and they are of fundamental importance in detecting incompressible surfaces in 3-manifolds. Character varieties are equally important in quantum topology and in mathematical physics, since the quantizations of character varieties of surface groups lead to quantum field theories and, in particular, quantum invariants of 3-manifolds and of links in them.

We will give a general introduction to character varieties and to some of their unintuitive anomalies. Then we will discuss the most elementary quantum deformations of character varieties called skein modules and skein algebras. We will survey their basic properties and their applications to other topics of quantum topology.

The course will be held on monday the 20th and tuesday the 21st of march with the following schedule :