7.2 Elefterios Soultanis: Curve fragments, plan-modulus duality, and differentiability of Lipschitz functions

Abstract: Rademacher's Theorem states that a Lipschitz function on Euclidean space is differentiable a.e. with respect to the Lebesgue measure. This statement is no longer true if we replace the Lebesgue measure with a singular measure. I will discuss a weaker form of differentiability, which remains true for singular measures, phrased in terms of curve fragments, i.e. (bi)-Lipschitz images of compact subsets of R. Key ingredients in such weak differentiability results are decompositions of measures into curve fragments, and their duality with modulus.