Title : Harmonic functions on Sierpinski carpets

Abstract : I will discuss a notion of Sobolev spaces and harmonic functions on Sierpinski carpets, which differs from the classical approach of potential theory in metric measure spaces. The goal is to define a notion that takes into account also the ambient space, where the carpet lives. As an application of carpet-harmonic functions we obtain a quasisymmetric uniformization result for Sierpinski carpets.

Title : A dichotomy for groupoid C*-algebras.

Abstract : Notions of paradoxical decompositions appear in the work of Hausdorff, Banach, and Tarski who showed that a discrete group satisfies the amenable/paradoxical divide. In this talk we study paradoxical phenomena in the field of operator algebras; directing our focus to C*-algebras arising from dynamical systems, graphs, and groupoids. Like Tarski, we use the type semigroup construction to move from non-paradoxicality to the existence of means or traces. These semi-groups witness the stably finite/purely infinite nature of the corresponding C*-algebras.

Title : Poincaré type inequalities on Hamming cube via martingale inequalities.

Abstract : Harmonic analysis is intimately related with martingale estimates.

But there is another type of discrete analysis, namely, harmonic analysis on Hamming cube (the math. foundation of Big Data science) that seemed to be disjoint from this relationship. We show how many classical (and some new) estimates on Hamming cube follow from martingale estimates. We also show why this is related to solving certain non-linear PDE of Monge–Ampère type and with classical inequalities in Gaussian spaces.

David Webb will present a new notion of effective dimension, inescapable dimension, which is in a sense dual to complex packing dimension.

The latter was introduced by Freer and Kjos-Hanssen in 2013 in the context of trying to show that the reals of effective Hausdorff dimension 1 are not Medvedev above the bi-immune sets.

Webb will show that the two notions are incomparable, among other results.

Speaker : Michael Yampolsky (University of Toronto)

Title : Computability of Julia sets.

Abstract : Informally speaking, a compact set in the plane is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. Julia sets are some of the best-known mathematical images, however, the questions of their computability and computational complexity are surprisingly subtle. I will survey joint results with M. Braverman and others on computability and complexity of Julia sets.