Minimal surfaces in spheres from random permutations

2024/06/25 Tue 15:00 - 16:30
Antoine Song
California Institute of Technology

Minimal surfaces are surfaces which locally minimize the area. They are 2d analogues of geodesics and have been intensively studied in Differential Geometry. I will define them and give a survey of what geometers have done and want to do with them. Next, I will discuss an unexpected relation between minimal surfaces in spheres, random matrices and the hyperbolic plane. Given two permutations on {1,...,N}, I will explain that one can (often) construct a minimal surface inside a round Euclidean sphere, whose area measures the "noncommutativity" of the permutations. The main result is that if the two permutations are picked uniformly at random, then the corresponding minimal surface is close to the hyperbolic plane with high probability. The proof combines classical results of Schoen-Yau, Sacks-Uhlenbeck for minimal surfaces, some representation theory, and a recent theorem of Bordenave-Collins for random permutations.