AWM-MAA Etta Z. Falconer Lecture 2016

Lecturer: Izabella Laba, University of British Columbia

Harmonic Analysis and Additive Combinatorics on Fractals

A plane is flat; a sphere is curved. Both are smooth, well behaved surfaces on which one can define measure and integration. If a harmonic analyst only knows the behaviour of analytic objects associated with a given surface, for example singular or oscillatory integrals, can she tell whether the surface is curved or flat? It turns out that, yes, the geometry of the surface is indeed reflected in such analytic estimates.

It might be somewhat surprising that similar phenomena have also been observed for fractals, including Cantor-type sets on the line. Some fractals behave, from the analytic point of view, as if they were flat; others display features typical of the sphere, and we have also seen additional types of behaviours that are never observed for smooth surfaces. The recent work investigating such phenomena highlights the connection to arithmetic properties of fractals, expressed in terms of “randomness” and “structure.”