This is a mathematical research project on core models and their fine structure, which is a subfield of set theory that was pioneered by Jensen. Core models are generalizations of Goedel's constructible universe, L, that arose in the the study of large cardinal axioms. The field today builds on the fundamental discoveries of Dodd, Jensen, Martin, Mitchell, Steel and Woodin. The project is on pure core model theory as well as its interactions with other areas, such as descriptive set theory, infinitary combinatorics and forcing. Several of the research problems are on combinatorial principles such as Diamond and Square, which were isolated by Jensen in his work on L. They are among the most commonly used tools in set theoretic constructions and have applications in neighboring fields such as model theory and topology. Other research problems have their origins outside of core model theory. These include open questions about forcing axioms, generic embeddings, singular cardinal combinatorics, canonical sets of reals and cardinal invariants of the powers of uncountable cardinals.
The axioms of set theory are the underlying assumptions of all of mathematics. Core model theory addresses some of the fundamental unresolved questions in set theory in direct and sophisticated ways. Beyond its foundational concerns, core model theory is connected to other areas, for example, cardinal arithmetic (the arithmetic of infinite numbers) and infinitary combinatorics (the combinatorics of infinite structures such as graphs and trees). The research problems addressed in this project lead to new core model theoretic techniques and applications.
