### The Banach-Tarski Paradox

Pure Mathematics Postgraduate Seminar

9th March 2018, 4:30 pm – 5:30 pm

Howard House, 4th Floor Seminar Room

One of the most counter-intuitive results in mathematics is the Banach-Tarski paradox. It says that a ball in R^3 can be broken into finitely many pieces which rearrange using rotations and translations to give two identical copies of the starting ball. The pieces used are constructed using the Axiom of Choice and are not Lebesgue measurable.

I will give a general definition of paradoxes, what B-T means for the existence of certain measures and describe some components of the proof that centre on a paradoxical decomposition of F_2, the free group with two generators. Some surface level intuition of measures might be useful, otherwise I will assume little background knowledge.

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