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 ended March 31, 2023. 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

  1. Outward compactness
    (with P. Holy and P. Lücke)

    Submitted. PDF. arXiv. Bibtex.

  2. Sigma_1-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders
    (with P. Lücke)

    Forum of Mathematics, Sigma. Volume 11, 2023. e103.
    DOI: 10.1017/fms.2023.102. PDF. arXiv. Bibtex.

  3. An undecidable extension of Morley's theorem on the number of countable models
    (with C. J. Eagle, C. Hamel, and F. D. Tall)

    Annals of Pure and Applied Logic. Volume 174, Issue 9, October-November 2023. 103317.
    DOI: 10.1016/j.apal.2023.103317. PDF. arXiv. Bibtex.

  4. Chang models over derived models with supercompact measures
    (with T. Gappo and G. Sargsyan)

    Submitted. PDF. arXiv. Bibtex.

  5. Towards a generic absoluteness theorem for Chang models
    (with G. Sargsyan)

    Submitted. PDF. arXiv. Bibtex.

  6. Uniformization and Internal Absoluteness
    (with P. Schlicht)

    Proc. Amer. Math. Soc. Volume 151, 2023. Pages 3089-3102.
    DOI: 10.1090/proc/16155. PDF. arXiv. Bibtex.

  7. Perfect Subtree Property for Weakly Compact Cardinals
    (with Y. Hayut)

    Israel Journal of Mathematics. Volume 253, March 2023. Pages 865-886.
    DOI: 10.1007/s11856-022-2385-4. PDF. arXiv. Bibtex.

  8. Determinacy and Large Cardinals

    Logic and Its Applications. Lecture Notes in Computer Science. Volume 13963.
    PDF. Bibtex.

  9. Constructing Wadge classes
    (with R. Carroy and A. Medini)

    Bulletin of Symbolic Logic. Volume 28, Issue 2, June 2022. Pages 207-257.
    DOI: 10.1017/bsl.2022.7. PDF. arXiv. Bibtex.

  10. The consistency strength of determinacy when all sets are universally Baire

    Submitted. PDF. arXiv. Bibtex.