Compactifications of character varieties and higher Teichmüller spaces

This course consists of three parts.

In the first part we develop the geometric invariant theory over the reals, in a self contained way. The goal is to show that the set of closed orbits of a linear action of a real reductive algebraic group on a real vector space has the structure of a semi algebraic set.

The second part is an introduction to the real spectrum of a commutative ring and the study of its basic topological properties. With this at hand we will define and study the real spectrum compactification of a semi algebraic set.

In the third part, the piece de resistance, we apply part one and two to the study of the real spectrum compactification of character varieties. We will characterize closed points of this compactification in terms of actions on buildings, and use this to establish a relation with the Weyl chamber length compactification of said character varieties. Finally we apply this theory to the real spectrum compactification of Hitchin components and establish a relationship with compactifications of Hitchin components by geodesic currents.

Prerequisites for this course is some basic geometric understanding of symmetric spaces of non compact type. We will reduce the prerequisites from real algebraic geometry to a few user friendly black boxes.