Boundary Phenomena: Harmonic Measure, Unique Continuation and Area Minimizing Currents

Objective

The study of harmonic functions is a cornerstone of partial differential equations. This project pursues three connected threads related to boundary behaviour: (i) harmonic measure, where we study its dimension and quantitative stability; (ii) boundary unique continuation for harmonic functions, extending Bers’ theorem in the plane to elliptic operators and improving unique continuation results for Neumann problems in rough domains; and (iii) boundary regularity for area-minimizing currents, in both the real-analytic setting (towards White’s conjecture) and the smooth case. These threads are closely connected through multiscale oscillation and propagation of smallness phenomena, and progress on them would mark significant advances and open new directions in analysis, geometry, and PDE.