The regularity of SLE

Abstract: I will present a quick introduction to SLE (Schramm-Loewner evolution), an important object appearing in random planar geometry and statistical physics. It describes a family of random fractal curves that exhibit conformal symmetries which underlie their prominence. Natural questions for fractal curves include their Hölder and p-variation exponents. I will explain how to find these and how to go further into finding the exact regularities (i.e. the correct logarithmic refinements). An important notion in the analysis is the "natural length" of the SLE curves. Our results (obtained jointly with Nina Holden) imply some important properties of this natural length, including a new characterisation.