Jonah Gaster (McGill University)

Title: Computing discrete harmonic maps between hyperbolic surfaces.

Abstract: Celebrated work of Eels-Sampson and Hartman asserts the existence of a harmonic diffeomorphism in any homotopy class of maps between a pair of homeomorphic compact hyperbolic surfaces. The study of harmonic maps has since been vastly generalized in the work of Gromov-Schoen, Korevaar-Schoen, and Jost, with wide-ranging application. Motivated by the nonabelian Hodge correspondence, we pursue a discrete version of the theory that is more accessible computationally; in particular our main theorem is that, in the setting of compact hyperbolic surfaces, the discrete energy functional is strongly convex (a property not known for the smooth energy functional). These ideas are implemented in a user-friendly computer program that I will present. This is joint work with Brice Loustau and Léonard Monsaingeon.