Long games and determinacy when sets are universally Baire

What do we mean, when we say that something is infinite? How many different infinities are there and how do they look like? These and similar questions form the fundamental pillars of set theory, a specialization of mathematical logic. The project Long games and determinacy when sets are universally Baire is located in this area, more specifically in the subarea called inner model theory. It sits at the boundary of what can proved in mathematics and aims for a better understanding of specific infinitely large objects (so-called large cardinals). Two central notions in inner model theory are large cardinals and determinacy axioms. They are of particular importance as at a first glance as well as historically they do not have much in common. But surprisingly it was shown in the 80s that these two notions have a deep connection. Large cardinals are axioms postulating the existence of unimaginably large numbers with useful properties. Determinacy axioms have a direct impact on the structure of sets of reals, i.e., on comparatively small objects in the hierarchy of infinities. They are relatively easy to define und postulate that in certain infinite two-player-games one of the players has a winning strategy. The fact that such an easily definable statement can neither be proven nor disproven makes the notion of determinacy particularly interesting. The concrete aim of this research project is to take our current understanding of the connection between large cardinals and determinacy axioms to a new level. The results could then lead to a better understanding of our mathematical universe. In addition, they could perspectively be used to transfer known theories from one area of set theory to another one.

What do we mean, when we say that something is infinite? How many different infinities are there and how do they look like? These and similar questions form the fundamental pillars of set theory, a specialization of mathematical logic. The project Long games and determinacy when sets are universally Baire is located in this area, more specifically in the subarea called inner model theory. It sits at the boundary of what can proved in mathematics and aims for a better understanding of specific infinitely large objects (so-called large cardinals). Two central notions in inner model theory are large cardinals and determinacy axioms. They are of particular importance as at a first glance as well as historically they do not have much in common. But surprisingly it was shown in the 80s that these two notions have a deep connection. Large cardinals are axioms postulating the existence of unimaginably large numbers with useful properties. Determinacy axioms have a direct impact on the structure of sets of reals, i.e., on comparatively small objects in the hierarchy of infinities. They are relatively easy to define und postulate that in certain infinite two-player-games one of the players has a winning strategy. The fact that such an easily definable statement can neither be proven nor disproven makes the notion of determinacy particularly interesting. In this research project the connection between large cardinals and determinacy was shown at a new level. More precisely, it was shown that the existence of so-called strong and Woodin cardinals is equally strong as the axiom of determinacy in a setting where all sets of reals are universally Baire. The answers an approximately ten years old question of Grigor Sargsyan.