Georgia Institute of Technology College of Sciences

On a given Riemann surface, we may talk about various projective structures that are compatible with the given complex structure. These structures are equivalent to what are known as $sl_2$ opers, or in terms of Faltings’ paper that we mainly follow, as permissible connections. Then there are several interesting questions we may ask about the monodromies of these objects, and the one we focus on today is whether the monodromy group is conjugate into $PSL_2(\mathbb R)$ in $PSL_2(\mathbb C)$ (“the real monodromy property”). In this talk, we illustrate (literally!) Faltings’ idea on this matter by some pictures and examples, and explain how uniformization and accessory problem fit into this theory. If time permitted, we will also see how this shows up in the recent Analytic Langlands program proposed by Etingof, Frenkel and Kazhdan.