# L'ORÉAL Austria Fellowship

Awarded in collaboration with the Austrian UNESCO Commission and the Austrian Academy of Sciences, with financial support of the Federal Ministry of Education, Science and Research, to four fellows per year in the fields of medicine, the natural sciences or mathematics. More details about the fellowship can be found on the webpage of the Austrian UNESCO Commission and in the announcement of the fellows on the ÖAW webpage. See also the announcement by the Faculty of Mathematics and the University of Vienna.

Portrait of myself and the fellowship (in German)

This project started on August 15, 2020 and ended on November 8, 2021. The awarded grants are 25,000€.

### Determinacy and Large Cardinals

The standard axioms of set theory, $\operatorname{ZFC}$, do not give a complete characterization of the structure of the sets of real numbers. In particular regularity properties such as Lebesgue measurability or the Baire property for rather simple (for example projective) sets of reals are not decided in $\operatorname{ZFC}$. A major part of set theory involves studying various extensions of $\operatorname{ZFC}$ and their properties to attack this problem. Therefore one of the main goals of research in set theory can be phrased as the search for the "right" axioms for mathematics.
Determinacy assumptions are natural extensions of $\operatorname{ZFC}$ postulating the existence of winning strategies in natural two-player games. They are known to imply for example certain regularity properties and supplement sets of real numbers with a lot of canonical structure. Other natural and well-studied extensions of $\operatorname{ZFC}$ are given by the hierarchy of large cardinal axioms. These extensions of $\operatorname{ZFC}$ are widely used and have many fruitful consequences in set theory and even in other areas of mathematics. Therefore understanding the connections between determinacy assumptions and the large cardinal hierarchy is of vital importance for answering questions which are left open by $\operatorname{ZFC}$ alone. My research in this L'ORÉAL Austria Fellowship is closely related to this overall goal.

### Publications submitted or published within this project

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

Israel Journal of Mathematics. Appeared online.
DOI: 10.1007/s11856-022-2385-4. PDF. arXiv. Bibtex.

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

Submitted. PDF. arXiv. Bibtex.

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

Accepted for publication in the Proceedings of the AMS. PDF. arXiv. Bibtex.

4. HOD in inner models with Woodin cardinals
(with G. Sargsyan)

The Journal of Symbolic Logic. Volume 86, Issue 3, September 2021. Pages 871-896.
DOI: 10.1017/jsl.2021.61. PDF. arXiv. Bibtex.

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

Submitted. PDF. arXiv. Bibtex.

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

Submitted. PDF. arXiv. Bibtex.

7. Closure properties of measurable ultrapowers
(with P. Lücke)

The Journal of Symbolic Logic. Volume 86, Issue 2, June 2021. Pages 762-784.
DOI: 10.1017/jsl.2021.29. PDF. arXiv. Bibtex.

8. Infinite decreasing chains in the Mitchell order
(with O. Ben-Neria)

Archive for Mathematical Logic. Volume 60, March 2021. Pages 771-781.
DOI: 10.1007/s00153-021-00762-x. PDF. arXiv. Bibtex.