Last week at my invitation, David Dumas spoke in the U Chicago geometry seminar and gave a wonderful introductory talk on the theory of convex real projective structures on surfaces. This is the first step on the road to what is colloquially known as “Higher Teichmüller theory”, and the talk made such an impression on me that I felt compelled to summarize it in a blog post, just to organize and clarify the material in my own mind.

Fix a surface (for convenience closed, oriented of genus at least 2). We are interested in the space of *convex real projective structures* on . This has at least 3 incarnations:

- it is a connected component of the character variety , the space of homomorphisms from into up to conjugacy (note: since all representations in this component are irreducible, one can really take the naive quotient by the conjugation action rather than the usual quotient in the sense of geometric invariant theory);
- it is topologically a cell, homeomorphic to , and can be given explicit coordinates (analogous to Fenchel-Nielsen coordinates for hyperbolic structures) in such a way that a coordinate describes an explicit method to build such a structure from simple pieces by gluing; and
- it has the natural structure of a complex variety; explicitly it is a bundle over the Teichmuller space of whose fiber is isomorphic to the vector space of cubic differentials on .

This last identification is quite remarkable and subtle, since is not a complex Lie group, and its action on the projective plane does not leave invariant a complex structure. Here one should think of as the space of (marked) *conformal structures* on , rather than as the space of (marked) hyperbolic structures on .

Following Dumas, we explain these three incarnations in turn. Some references for this material are Goldman, Benoist, and Loftin.