Speaker: Antoine Song (California Institute of Technology)
Time: 2025-11-14 09:00-11:00
Zoom Meeting ID:965 8558 2035 Passcode:454139
Recording: Recording_Shape of random harmonic maps into high-dimensional spheres.mp4
Abstract: Harmonic maps from Riemann surfaces to Riemannian manifolds are fundamental objects in geometry, generalizing harmonic functions. In general, it is very difficult to describe them explicitly or to classify them. In this talk, I will discuss some recent work where we study the shape of harmonic maps into spheres which are “random”. Surprisingly, we will see that the shape of such objects is often close to a hyperbolic plane, a well-known canonical shape in geometry. I will describe some of the main ideas needed to prove such a result and mention some applications and open questions. This result was recently used to settle an open problem of S.T. Yau on the existence of negatively curved minimal surfaces in Euclidean spheres.
About Speaker:
Antoine Song is a professor at California Institute of Technology. He is also a Sloan research fellow. Previously, he completed my PhD in 2019 at Princeton University, continued as a postdoc at UC Berkeley, and was a Clay fellow from 2019 to 2024.
His research is mostly in Differential Geometry and Geometric Analysis. He is broadly interested in shapes, especially those which are optimal under natural constraints. In particular, many of his papers involve minimal surfaces and harmonic maps. Recently, he has started to explore their properties in high-dimensional spaces, and new connections with other fields like representation theory, geometric group theory, random matrices.
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