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

在强公理存在的情况下，研究者将继续研究连续体假设的可定义反例的存在性，目的是证明它们的不存在。这已经在组合集合论和描述集合论中产生了丰厚的红利。他还将继续将顺从和非顺从的群体行为与集合论联系起来的工作。此外，他还将尝试将他以前对著名的关于Banach-Tarski悖论的“Marczewski问题”的解决方案推广到波兰空间中的任意非服从同胚群。这将使用具有Baire性质的集合来精确地刻画波兰空间的矛盾分解。他还将研究新发现的逻辑和集合论与拓扑动力学和遍历理论之间的联系。一个著名的测度论悖论，Banach-Tarski悖论表明，关于三维空间中集合体积的普通直觉是不可靠的。特别是，如果允许集合足够奇怪，当然不是任何可以绘制或容易想象的东西，那么三维空间中的一个实心球可以被解剖成有限个集合，这些集合可以重新排列和重新组装成两个实心球，每个都与原始的一个相同。一个曾多次尝试的老问题是，在这种矛盾的球分解中涉及的几盘是否至少具有最低程度的规则，即所谓的拜尔性质。具有Baire性质的不可测集虽然不容易绘制或想象，但在经典分析中确实会出现，因此不会被认为与实际相距太远，以至于很容易被完全忽略。这位调查员和一位同事驳斥了人们普遍认为这种分解是不可能的怀疑，开发了一些技术机制来处理这个问题和相关问题，并证明了这种分解确实存在。本课程还将讨论纯集合论及其应用的其他主题。

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.