Generic large cardinals and determinacy

Set theory is the mathematical study of infinity and has been very successful both in answering its own deep fundamental questions and in being applied elsewhere. Gödels Program is a major set theoretic undertaking addressing the most fundamental set theoretic issue: the fact that many basic questions about infinite sets are not decided by the standard axiom system ZFC. One important aspect of Gödels Program is to investigate theories of high logical strength that may disagree with each other on some set theoretic questions but can be seen as different aspects of the same endeavor through their mutual interpretability. In this project, our goal is to establish bridges between two such theories: generic large cardinals and strong forms of the Axiom of Determinacy. One of the prime objectives of this project is the construction of dense ideals on the first omega uncountable cardinals. We aim to construct such ideals using Cohens method of forcing over both models of determinacy as well as models of traditional large cardinals, which will show that the theory of dense ideals can be interpreted by two different foundational frameworks. Such ideals have already been constructed on the first two uncountable cardinals, but the construction of such ideals on higher cardinals has been a major open problem in set theory. The existence of such ideals has number of applications in infinitary combinatorics, model theory, and algebra. We also aim to calibrate the exact consistency strength of the theory asserting the existence of such ideals by constructing models of determinacy from such ideals, providing the desired mutual interpretation. We expect this project to be highly innovative, as the potential connections between determinacy and dense ideals above the first uncountable cardinal have not yet been explored. Our methods will involve innovations in forcing technology, both in the classical context of ZFC with large cardinals, as well as in the more bespoke context of forcing over models of determinacy. Our consistency calibration pushes against recently-discovered limitations of the core model induction, and thus should lead to breakthroughs in this methodology.