Mathematical Sciences: Combinatorial Set Theory and Forcing

鲍姆加特纳教授将致力于组合集合论，研究与强迫理论和一般集理论有关的问题。研究领域包括：涉及假设CH的omega-3性质的划分演算中的问题，集中在ZFC CH中可证明的性质及其必然伴随的独立性问题；关于划分序数、极化划分关系和集合的划分关系的相关问题；关于闭无界集的大小问题；关于“基数不变量”的问题，既与测度论、范式论有关，又与二次空间理论有关。集合论是所有数学的基础。重要的问题涉及所有数制、代数、几何和分析背后的公理，某些公理是必要的吗？在某些情况下，它们是否无关紧要？在这方面特别重要的是连续统假说(CH)。对它在数学中的作用的理解启发了20世纪一些最深刻的研究。

Professor Baumgartner will work in combinatorial set theory, on questions connected to the theory of forcing and generic sets. Areas of research include problems in the partition calculus involving properties of omega-3 assuming CH, focusing on properties provable in ZFC+CH together with the necessarily attendant independence questions; related problems on partition ordinals, polarized partition relations and partition relations for sets; problems on the sizes of closed unbounded sets; and problems on "cardinal invariants" connected both with measure and category theory and with the theory of quadratic spaces. Set theory lies at the foundation of all mathematics. Important questions concern the axioms that underly all number systems, algebra, geometry and analysis, Are certain axioms necessary? Are they irrelevant in certain contexts? Particularly significant in this regard is the Continuum Hypothesis (CH). Understanding its role in mathematics has inspired some of the most profound investigations of the 20th century.