Set Theory in Topology and Dynamics

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Topological spaces are abstract mathematical structures whose geometric properties are not affected by the continuous change of shape or size. From early on, set theory has played a crucial part in study of topological spaces. The relationship between set theory and topology has grown deeper over time. This project concerns the ongoing use of set-theoretic methods in topology and topological dynamics. The project also provides research training opportunities for graduate students.The set-theoretic methods forming the focal point of this project are various manifestations of the unifying theme of elementary submodels. Included in this theme are so-called "L-like" principles, like Jensen's square and its variants, that arise from considerations of elementarity and reflection in inner models. Making use of such principles in topology typically requires the assumption that the set-theoretic universe be close to Gödel's constructible universe L in some way. This project aims to explore the topological consequences of these L-like principles, as well as the sometimes exotic possibilities that can exist in their absence. The latter entails a topologically-minded investigation of large cardinals, forcing models built from large cardinals, and the singular cardinal combinatorics arising therein.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.