Mathematical Sciences: Combinatorial Set Theory and Forcing

鲍姆加特纳教授将研究组合集理论，研究与强迫理论和一般集相关的问题。研究领域包括在分区演算问题，涉及ω -3的性质假设CH，重点是在ZFC+CH中可证明的性质以及必然伴随的独立性问题；关于划分序数、极化划分关系和集的划分关系的相关问题关于闭无界集的大小问题并讨论了与测度论、范畴论和二次空间理论有关的“基数不变量”问题。集合论是所有数学的基础。重要的问题涉及所有数字系统、代数、几何和分析背后的公理。某些公理是必要的吗？它们在特定的语境中是否不相关？在这方面特别重要的是连续统假设（CH）。理解它在数学中的作用激发了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.