Orateur
Description
I will discuss three interrelated problems about Riemann surfaces:
How large can the spectral gap of the Laplacian be? (Old, e.g. Selberg)
Do there exist negatively curved minimal surfaces in Euclidean spheres? (Yau)
How good (in terms of conformal factors) can holomorphic immersions of Riemann surfaces in flat complex manifolds be?
Progress on all these questions comes from understanding notions of convergence of unitary representations of both PSL_2(R) and the fundamental groups of closed surfaces. I'll give an introduction to (induced) representations of PSL_2(R), and an introduction to the notions of convergence we need.
To give attributions, on question 1 I will discuss my results with Hide and Louder but perhaps not with the same proofs.
On question 2 I report on the works of Song and Ancona, Labourie, Roig-Sanchis, and Toulisse.
The last topic is a variation of a theme of Kazhdan, Rhodes, and McMullen, from a new perspective, which raises some interesting questions about group C*-algebras.