The PI is investigating several aspects of large cardinal theory in the region of Woodin cardinals, including (1) Long Games; (2) Iterability; and (3) connections with Descriptive Set Theory. 1 - Results obtained by the PI through previous NSF support demonstrate the tight connection between Woodin cardinals and games of variable countable length. The existence of iterable models for specific large cardinals can be used to yield the determinacy of games of specific lengths, with increasing length corresponding to increasing large cardinal strength. These games in turn can be used to decide statements over minimal inner models for large cardinals. The PI is attempting to extend this correspondence, the first main test case being open games of length omega one. 2 - The study of Woodin cardinals leads naturally to a transfinite game known as the iteration game. Winning strategies ("iteration strategies") for the good player in this game are essential to the comparison process that lies at the heart of inner model theory. Proving that iteration strategies exist is the single most important problem in the field. The PI has been working on this problem, contributing to the existing pool of partial results. Several problems, related to the general problem of iterability but specialized and more concrete, are investigated as part of this project. This investigation should provide increased understanding of the main problem, and hopefully lead to additional partial results. 3 - The PI is working to solidify the connections between large cardinals and definable sets of real numbers. Extensive work, much of it done during the '70s and '80s, provides a detailed analysis of definable sets of reals assuming determinacy. Later work obtained determinacy from large cardinals. The PI is working to emulate the existing analysis of definable sets of reals, working directly from large cardinals. The point here is to try to convert methods used in the study of definable sets of reals, into methods which can be used in inner model theory and the study of large cardinals.
Set Theory is a branch of Mathematics which attempts to understand the universe of Mathematics, that is the collection of all objects studied by Mathematicians, or more precisely the collection of all _sets_. (Numbers, groups, functions, etc., can all be represented as sets.) Set Theorists view this universe as a structure in its own right, a structure which can itself be analyzed using mathematical reasoning. For example, a Set Theorist may ask "is there an embedding of the universe into a similar, yet not identical, structure?" Set Theorists work with such embeddings ("elementary embeddings") in much the same way that one would work with functions on numbers, asking "how similar is the target structure to the original structure?" and "what's the smallest size of a set actually moved by the embedding?" Such questions about embeddings of the universe form a subfield of Set Theory known as the study of large cardinals. The terminology here hints at the answer to the last question mentioned. Indeed, the sets actually moved by elementary embeddings are substantially larger than any object studied in other branches of Mathematics (substantially larger than the set of real numbers for example). Yet it turns out that the existence of elementary embeddings has concrete effects on lower level objects, including concrete effects on real numbers. This project is part of the on-going effort to better understand large cardinals, and better understand their effects on the line of real numbers. The PI, like other researchers in the field, is motivated by the connection between the abstract and the concrete.
