MVEX01-19-29 The Banach-Tarski paradox

​The Banach-Tarski paradox is often stated as follows: Given a solid ball in three dimensions it is possible to cut it into a finite number of pieces and rearrange these pieces to make two balls, each exactly the same as the original ball.

The result was proved by Banach and Tarski (1924), building on work of Hausdorff (1914). The paradox, along with the fact that no such paradox exists in one or two dimensions, hints at the subtle nature of the concept of volume, as well as deep properties of the group of translations and rotations of three dimensional space.

The project will begin by developing the basic theory of free groups, and understanding Lebesgue measure on Euclidean space. By introducing the idea of a paradoxical decomposition we will be able to prove the Hausdorff paradox, then build up to proving the Banach-Tarski paradox. From this point our investigation may take two directions depending on which question is of more interest: What exactly is it about the group of reflections and rotations of three dimensional space that gives rise to the paradox? Is there an alternative notion of volume that does not allow the paradox to exist?

Projektkod MVEX01-19-29
Gruppstorlek 2-4 studenter
Målgrupp GU- och Chalmersstudenter. För GU-studenter räknas projektet som ett projekt i Matematik (MMG900/MMG910).
Projektspecifika förkunskapskrav Basic group theory.
Se respektive kursplan för allmänna förkunskapskrav. Utöver de allmänna förkunskapskraven i MVEX01 ska Chalmersstudenter ha avklarat kurser i en- och flervariabelanalys och linjär algebra.
Handledare Andrew McKee, andrew@chalmers.se
Examinator Maria Roginskaya, Ulla Dinger
Institution Matematiska vetenskaper

Publicerad: fr 02 nov 2018.