Forcing, Large Cardinals, and Infinitary Combinatorics

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This program of research concerns combinatorial set theory, which is the study of certain infinite mathematical structures, including graphs, trees, and linear and partial orders. These structures are ubiquitous in mathematics and can be studied from many points of view. This project aims to help answer fundamental mathematical questions by extending understanding of the cardinality of these structures. The research involves ideas from areas including logic, algebra, and the theory of infinite games.The main topics in the project are compactness properties, singular cardinal combinatorics, and definability. The goals include producing models where many consecutive cardinals simultaneously have the tree property, getting refined information about singular cardinals through the study of their cardinal invariants, and constructing novel extender-based forcings. The tools will include inner model theory, forcing, and possible cofinalities (PCF) theory.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.