Thomas Wolff Memorial Lecture in Mathematics
Abstract: Rademacher's Theorem states that a Lipschitz mapping from one Euclidean space to another is differentiable except on a set of (Lebesgue) measure zero. In this lecture and the next I will discuss joint work with Marianna Csornyei: Given a set E of measure zero, there is a map from Euclidean space to itself that is not differentiable on E. This is an old and easy result in dimension D = 1. In two dimensions this result was proven by G. Alberti, M. Csornyei, and D. Price. Their proof has two parts. First there is a combinatorial argument (special to dimension = 2), and a "covering argument", which works in any dimension. In this will explain some of the technical tools that go into the work of Csornyei and PJ. Critical here is the notion of the concept of "Tangent Cones" for sets of measure = 0. I will also review some basic machinery from harmonic analysis that is used in our proof.