Large cardinals and small sets

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Larson proposes work on five related topics, all centering around the extent to which the absoluteness results and detailed analysis of inner models of determinacy in the context of large cardinals can be lifted to larger models satisfying larger fragments of the Axiom of Choice. One topic is the so-called universally measurable sets, where the fundamental open question is whether all such sets can have the property of Baire. One topic concerns the fragment of the Axiom of Choice which holds when a model of determinacy is expanded by adding a Ramsey ultrafilter. Another concerns lifting to the third uncountable cardinal Woodin's results on getting forcing axioms at the second uncountable cardinal by forcing over a determinacy model. The fourth concerns application of techniques from set theory to Shelah's study of Abstract Elementary Classes. Finally, Larson proposes to study ideals on the first uncountable cardinal from the point of view of certain inner models satisfying a small fragment of Choice. The standard axioms for set theory, the Zermelo-Fraenkel axioms, serve as the commonly accepted foundations for mathematics, though as mathematics becomes more abstract, more and more issues arise which are not resolved by these axioms. Many of these issues are studied by set theorists, in hope of finding the right extension of these axioms. Developments in the study of such possible extensions have had a dramatic foundational impact in the last thirty years, affecting many areas of mathematics, and even philosophy. The PI works on the border between some of the more technical, inward-directed areas of set theory and more classical areas with connections to other fields. Much of his work consists of finding applications of these more technical areas, and exposing them to a wider audience.