“Lipschitz functions and the geometry of rough sets”
Martedì 29 Settembre 2026, ore 13:30 - Aula 1BC450 - Andrea Merlo (Universidad del Pais Vasco)
Abstract
Rademacher’s theorem tells us that Lipschitz functions on Euclidean space admit linear approximations at almost every point. Can the approximation properties of Lipschitz functions also tell us whether a set has a regular geometric structure? For sets whose measure is comparable to a fixed power of the radius, David and Semmes proposed the following principle: if every Lipschitz function can be well approximated by affine functions, with quantitative control of the locations and scales where approximation fails, then the set must be uniformly rectifiable. This means that every ball contains a substantial portion of the set that admits a Lipschitz parametrization, with uniform bounds. This is the WALA conjecture. In this talk, I present a proof of the conjecture and explain the geometric ideas connecting curves, differentiability and affine approximation. I also briefly discuss applications to elliptic partial differential equations, arising from joint work with Mihalis Mourgoglou, where estimates for solutions reveal geometric properties of the boundary.
