Combinatorial Set Theory, Model Theory of Abstract Elementary Classes, and Borel Combinatorics

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This project explores questions in three separate areas of logic and foundations of mathematics. First, the project will study possible behaviors of the mathematical universe using Zermelo-Frenkel set theory as a foundation. The main goal is to understand infinitary combinatorics through the lens of forcing and large cardinals. Second, the project will investigate the connections between set theory and an abstract view of model theory. Many assertions in model theory turn out to have set theoretic content, and the investigator will pursue the question of when there is such a connection and when there is not. Third, the project will explore the relatively recent field of descriptive graph combinatorics and its connections with classical questions in geometry and measure theory. From infinitary combinatorics, the project will focus on the tree property, which is a compactness principle that abstracts the Konig infinity lemma to higher cardinals. A specific goal of the project is to explore the conjecture that modulo the consistency of large cardinals the least uncountable cardinal is the only cardinal which provably carries a counterexample to the tree property. The project will also pursue other similar questions. From model theory, the investigator will study model theoretic notions connected to the classification theory of abstract elementary classes and their set theoretic content. Particular properties of interest are tameness and locality, categoricity, and Hanf numbers. From descriptive graph combinatorics, the investigator will explore matchings and colorings in definable graphs. This study has already been shown to have interesting connections with questions in geometry, for example Marczewski's question about whether the Banach-Tarski paradox is possible using Baire measurable pieces.