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

调查人员将继续研究是否存在 可定义的反例，以连续统假设在 强公理的存在，目的是证明他们的非- 存在 这已经在两个方面产生了丰厚的红利 组合和描述集合论。 他还将继续 将服从和不服从的群体行动与集合联系起来的工作 理论 此外，他将尝试扩展他以前的解决方案， 到著名的关于巴拿赫-塔斯基悖论的“马切夫斯基问题”， 波兰空间的同胚的任意非顺从群。 这将提供一个确切的描述， 波兰空间的分解，使用具有以下性质的集合 贝尔 他还将调查新发现的联系 逻辑和集合论之间的联系 动力学和遍历理论。 一个著名的测度论悖论，巴拿赫-塔斯基悖论， 显示了关于三个集合的体积的普通直觉- 空间是不可靠的。 特别是，如果集合是 允许足够怪异，当然不是任何可以 可以画出或很容易想象，那么一个三维空间中的实心球可以是 被分解成有限数量的集合，这些集合可以是 重新排列组合成两个实心球 到原来的那个。 一个古老的问题， 尝试是否涉及这样一个矛盾的集合 球的分解可以具有至少最小程度的 规则性被称为Baire属性。 虽然不容易 画或想象，不可测集的性质Baire做 出现在经典分析中，因此并不被认为是如此遥远 很容易被完全忽略。 研究者 和一位同事否认了人们普遍的怀疑， 分解是不可能的，开发了一些技术机器 处理这个问题和相关的问题，并表明， 分解确实存在。 纯集合论中的其他主题 并将考虑其应用。

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.