CHANGE OF DATE Joint Colloquium: Hansen -- Littlewood’s one circle problem and Liouville’s theorem

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.