Miklós Laczkovich:
The Banach–Tarski paradox
The Banach-Tarski paradox is one of the most surprising results of pure mathematics. It states that a three dimensional ball can be decomposed into a finite number of pieces such that a suitable rearrangement of the pieces constitutes a decomposition of a larger ball, or, more generally, of an arbitrary bounded set with a nonempty interior. In this course we show how the result emerged from the problem of invariant measures, and cover the preliminaries needed for the proof including some geometry (isometries of the Euclidean space), and group theory (free groups). Then we prove the Banach-Tarski paradox, and discuss some improvements and generalizations.