Mathematical Sciences: Some Investigation into the ContinuumHypotheses and also Set-Theoretic Aspects of Group Actions

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The investigator will continue work on the existence of definable counterexamples to the continuum hypothesis in the presence of strong axioms, the aim being to prove their non- existence. This has already yielded rich dividends in both combinatorial and descriptive set theory. He will also continue work connecting amenable and non-amenable group actions with set theory. Furthermore, he will try to extend his previous solution to the famous "Marczewski Problem" about Banach-Tarski paradoxes to arbitrary non-amenable groups of homeomorphisms of Polish spaces. This will provide an exact characterization of paradoxical decompositions of Polish spaces using sets with the property of Baire. He will also investigate newly discovered connections between logic and set theory on the one hand and topological dynamics and ergodic theory on the other. A famous paradox of measure theory, the Banach-Tarski paradox, shows that ordinary intuition about volumes of sets in three- dimensional space is unreliable. In particular, if the sets are allowed to be sufficiently bizarre, certainly not anything that can be drawn or easily imagined, then a solid ball in 3-space can be dissected into a finite number of sets, and these sets can be rearranged and reassembled to form two solid balls, each congruent to the original one. An old question that had resisted numerous attempts was whether the sets involved in such a paradoxical decomposition of the ball could have at least a minimal degree of regularity known as the property of Baire. Although not easy to draw or imagine, non-measurable sets with the property of Baire do arise in classical analysis and so are not considered so remote from reality as to be easy to ignore altogether. The investigator and a co-worker have defied widely held suspicions that such decompositions are impossible, developed some technical machinery for treating this and related questions, and shown that such decompositions do indeed exist. Other topics in pure set theory and its applications will be considered.