6 February 2025 (Thursday) 16:00 - 17:15  /  129-301

7 February 2025 (Friday) 16:00 - 17:15  /  129-301

10 February 2025 (Monday) 16:00 - 17:15  /  129-301

12 February 2025 (Wednesday) 16:00 - 17:15  /  129-301

Title: Fock-Goncharov coordinates and spectral networks

Abstract: In 2006 V. Fock and A. Goncharov introduced moduli spaces generalizing the (decorated) Teichmüller space for punctured surfaces. These moduli spaces are built upon the character varieties for higher-rank split real Lie groups (though we will focus in this lecture on PGL_n/SL_n). The theory they developed revolves around building so-called "cluster coordinates" on these moduli spaces that satisfy very strong algebraic and combinatorial properties. One of the outcome of their work is the construction of the first examples of what is now called "higher Teichmüller spaces". In the first part of this mini-course, I will give an introduction to these moduli spaces and their cluster coordinates. In the second part, I will draw the link between the theory of Fock-Goncharov and another type of object called "spectral networks", introduced by D. Gaiotto, G. Moore and A. Neitzke, bringing a more geometric/topological aspect to the story.