Max Goering

University of Washington

Talk Title: Non-local curvatures and the geometry of measures

Abstract: Uniformly rectifiable sets have played a fundamental role in the development of harmonic analysis in on-smooth settings. In one-dimension the 1996 paper of Mattila, Melnikov, and Verdera first made use of the classical Menger curvature to provide a new proof that Uniformly Rectifiable (UR) curves are characterized by the $L^{2}$-boundedness of the Cauchy integral operator in the plane. This opened the floodages in relating the analytic and geometric properties of one-dimensional sets and measures. Higher-dimensional analogs of the Menger curvature were much sought after until Farag showed that there was no algebraic generalization of the Menger curvature which could relate to the $L^{2}$-boundedness of the Riesz kernels. Nonetheless, Lerman and Whitehouse proved that geometrically motivated generalizations can be used to characterize uniformly rectifiable sets in real-separable Hilbert spaces. In this talk, we discuss a new characterization of the wilder class of rectifiable measures in terms of these types of discrete curvatures.