In the first part of the talk I propose a gentle introduction to character varieties, the space of representations of the fundamental group of a surface. In particular I highlight motivations from algebra, geometry and physics.
In the second part I will show the link to flat connections and differential operators. In particular I'll present a generalization of the concept of opers, using a modified version of the WKB method.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/158