Georgia Institute of Technology College of Sciences

Say you’ve got an (orientable) surface S and you want to do geometry with it. Well, the complex plane C has dimension 2, so you might as well try to model S on C and see what happens. The objects you get from following this thought are called complex structures. It turns out that most surfaces have a rich but manageable amount of genuinely different complex structures. I’ll focus in this talk on how to think about comparing and deforming complex structures on S. I’ll explain the remarkable result that there are highly structured “best” maps between (marked) complex structures, and how this can be used to show the right space of complex structures on S is a finite-dimensional ball. This is known as Teichmüller’s theorem, and I’ll be following Bers’ proof.