Series: Penn State Logic Seminar Date: Tuesday, September 20, 2005 Time: 2:30 - 3:45 PM Place: 106 McAllister Building Speaker: John D. Clemens, Penn State, Mathematics Title: Continuous Banach-Tarski Paradoxes Abstract: In 1924, Banach and Tarski showed (using the Axiom of Choice) that a sphere is paradoxical, i.e. it can be partitioned into finitely many pieces and rearranged using isometries to form two spheres, each of equal volume to the first. De Groot asked whether this could be done continuously, that is, if the isometries could be extended to paths (continuous motions) so that the pieces remain disjoint at each instant. Recently, Trevor Wilson settled this question by proving the following general result: Any two bounded subsets of R^n (for n at least 2) which are equidecomposable using proper isometries are in fact continuously equidecomposable. I will explain these notions and present Wilson's result and some of its consequences.