FWF Elise Richter Project V844

Awarded to support highly qualified female scientists in all disciplines to enhance their university career. More details on the FWF Elise Richter Programme can be found on the webpage of the FWF and in the FWF project finder. See also the announcement by the Faculty of Mathematics.
This project started on February 22, 2021 and will run until February 21, 2025. The awarded grants are 341,754€.

### 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 80’s 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.

### Publications submitted or published within this project

Sigma_1-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

(with P. Lücke)Uniformization and Internal Absoluteness

(with P. Schlicht)An undecidable extension of Morley's theorem on the number of countable models

(with C. J. Eagle, C. Hamel, and F. D. Tall)The consistency strength of determinacy when all sets are universally Baire

Perfect Subtree Property for Weakly Compact Cardinals

(with Y. Hayut)Accepted for publication in the Israel Journal of Mathematics. PDF. arXiv. Bibtex.

Constructing Wadge classes

(with R. Carroy and A. Medini)Accepted for publication in the Bulletin of Symbolic Logic. PDF. arXiv. Bibtex.