Type: Seminar

Modified: Fri 28 Jan 2011 0905; Fri 28 Jan 2011 0910; Fri 28 Jan 2011 1753

Distribution: World

Expiry: 3 Feb 2011

Auth: athomas@p615.pc (assumed)

Update: the joint colloquium next week has been rescheduled to THURSDAY 3 February, so that the lunch beforehand does not clash with the meeting on Friday about ARC Discovery Projects. Other details remain the same. There will be a joint colloquium given by Prof. Wolfhard Hansen at the University of Sydney on Thursday 3 February. We will leave for lunch from the 2nd floor of Carslaw at 1pm. In case of any difficulties locating the lunch crowd, my mobile number is 0426 243 411. =========================================== THE UNIVERSITIES OF SYDNEY AND NEW SOUTH WALES SCHOOLS OF MATHEMATICS AND STATISTICS ___________________JOINT COLLOQUIUM______________________ Speaker: Prof. Wolfhard Hansen (Bielefeld) Date: Friday, 4 February 2011 Time: 14:30 Venue: Carslaw 175, University of Sydney Title: Littlewood’s one circle problem and Liouville’s theorem Abstract: Littlewood’s one circle problem was the question, if a continuous bounded function f on the open unit disk U is harmonic provided that, for every x in U, there exists 0<r(x)<1-|x| such that the average of f on the circle of radius r(x) centered at x is equal to f(x). It is known since 1994 that the answer is no. The original construction of a counterexample is very delicate. Still using a sophisticated random walk jumping on small annuli, it becomes simpler and more transparent by an application of a striking result due to M. Talagrand yielding a closed set of area zero in the plane which, for each point of the real line, contains a circle centered at this point. In contrast to the subtlety of the counterexample for the unit disk, there is an elementary proof showing that, if the unit disk is replaced by the plane and r(x)\le |x|+M for large values of |x|, then f is constant. The corresponding problems are open in dimension d\ge 3.