___________________JOINT COLLOQUIUM______________________ Speaker: Prof. Tuomas Hytönen, (University of Helsinki) Title: How to choose random cubes, and why? Date: Friday, August 7, 2009 Time: 2:00 pm Venue: Room 275 Carslaw Building, University of Sydney Abstract: A standard tool in mathematical analysis is the refining sequence of partitions of the Euclidean space into cubes with side-lengths equal to powers of two, the so-called dyadic cubes. Associated to the system of dyadic cubes, there is a natural basis of square-integrable functions, the Haar system, which has proven useful e.g. in the analysis of singular integral operators. In their work on singular integrals related to quite general measures, Nazarov, Treil and Volberg observed that, while certain matrix coefficients of such an operator with respect to the Haar basis do not admit any good control, such bad situations only occur "rarely", which can be made precise in the sense of probability by working with a randomly chosen dyadic system, instead of a fixed one. I will discuss how these random cubes a chosen in the original Euclidean framework, and also what they could be in a more general metric space. The Speaker will have been taken to lunch at the University of New South Wales before the colloquium.