Generic regularity of harmonic maps

AMS 2026 Fall Western Sectional Meeting, Arizona State University — November 2026

Abstract

In this talk, I will first introduce harmonic maps and review some basic aspects of their regularity theory. I will then discuss recent progress on the regularity of harmonic maps with generic boundary data. Roughly speaking, the result shows that, for generic boundary data, singularities can only be modeled on harmonic spheres of the lowest possible Morse index.

When the target is a \(k\)-sphere, combining this result with my work on the optimal regularity theory for harmonic maps into spheres yields a more explicit description of the generic singular set. For , \(k\leq6\) a generic harmonic map has only isolated singularities, each modeled on a rotation of the standard sphere. In contrast, for \(k\geq7\), generic harmonic maps have no isolated singularities.