Harmonic Maps in High-Dimensional Spheres, Representations and Random Matrices - 1/4

Explore harmonic maps from 2-dimensional surfaces to high-dimensional spheres through the lens of unitary representations of surface groups in this 57-minute lecture by Antoine Song from the California Institute of Technology. Delve into the intersection of geometric analysis, topology, and representation theory as you examine how geometric objects can be studied from topological data and vice versa. Investigate rigidity phenomena that govern the shape of harmonic maps into spheres, with particular focus on two key regimes: the high-dimensional asymptotic case where random matrix theory becomes crucial, and the infinite-dimensional scenario where representation theory of PSL2(R) takes center stage. Gain insights into this sophisticated mathematical framework that bridges differential geometry, algebraic topology, and modern analytical techniques, presented as part of a comprehensive course series at the Institut des Hautes Etudes Scientifiques.