Colloquium – Multifractal spectrum of planar harmonic measures

Speaker: Adi Glucksam

Abstract: Classical fractals are self-similar sets, namely sets that “look the same” at different scales. Multifractal sets are geometric objects where different patterns emerge along different sequences of scales. In nature, a snowflake is a fractal while coastlines form a good example of multifractal behaviour. Multifractal Analysis provides tools to analyze these more complex phenomena, which are no longer recursive in the classical sense.In this talk, we focus on Multifractal Analysis of harmonic measures on simply connected domains. A central problem is the connection between two spectra; the dimension of the set where the harmonic measure has a certain decay, and the dimension of the set where the distortion of the conformal map (the Riemann map from the disk to the domain) has a specific growth rate. This is where multifractality emerges – different scales present different behaviour decay or growth-rate.We will trace the history of this problem—exploring what was known, what was long believed to be true, and what was tacitly assumed in the literature. Finally, I will present a surprising new phenomenon that challenges these classical intuitions. This is a developing story, based on a joint work with I. Binder.