Samuel Dodds : Ruiziewicz problem on the sphere

Posted by Alexander Furman , part of the Graduate Groups and Dynamics Seminar.

At

Sept. 18, 2018, 3 p.m.

In

1227 SEO

Abstract

The Ruziewicz (a.k.a Banach-Ruziewicz) problem on the $n$-sphere $S^n$ is the question whether the Lebesgue measure is the <i>unique</i> rotation invariant normalized <i>mean</i> on the Lebesgue sigma-algebra of the sphere. Here a <i>mean</i> is a finitely additive probability measure. For $n=1,2$ the answer is negative; for $n\ge 5$ it is positive as was proved independently by Margulis and Sullivan. For $n=2,3$ the positive answer was proven by Driendfeld. In the talk the connection between the problem and an existence of a group $G<SO(n)$ acting with a spectral gap on $L^2_0(S^n)$ will be explained, and proven for $n\ge 5$.