Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions

This paper surveys different topics where the theory of quantitative rectifiability plays a central role. First, it reviews the characterization of rectifiability in terms of square functions involving $\beta$ type coefficients and the ${\epsilon }^2$ conjecture of Carleson. It also discusses the deep connections between rectifiability and the $L^2$ boundedness of Riesz transforms and their application to the Painlevé problem for Lipschitz harmonic functions. Finally, the paper explores recent major advances in connection with harmonic measure and the $L^p$ solvability of the Dirichlet, regularity, and Neumann problems for the Laplace equation in rough domains, emphasizing the key role of quantitative rectifiability in these developments.