This project consisted of a two-year outgoing phase at the University of California, Berkeley, and a one-year return phase at Technische Universität Wien.
The scientific results of this project were published in 10 peer-reviewed research articles. As of now, all of these articles have been submitted, with several of them already accepted for publication.
Additionally, the principal investigators gave 20 talks advertising the results obtained in this project to the scientific community.
The four highlight results obtained in the project are:
1. A characterization of the descriptive complexity of the set of models of a first-order theory in terms of the quantifier complexity of axiomatizations of the theory. This result, obtained with Andrews, Gonzalez, Lempp, and Zhu, shows that in order to get a grasp of the descriptive complexity of a theory's models, one just has to look at the quantifier complexity of the theory's simplest axiomatization, which is usually much easier to determine.
2. A result showing that elementary bi-embeddability is an analytic complete equivalence relation and, thus among the most complicated analytic equivalence relations. The PI showed this by reducing bi-embeddability to elementary bi-embeddability. Their reduction shows that the bi-embeddability relation and the elementary bi-embeddability relation not only have the same descriptive complexity but while their degree spectra are not the same, they are at most one jump apart. Intuitively, this means that the computational content of a structure relative to the bi-embeddability and the elementary bi-embeddability relation can be different, but that this difference is very small.
3. An exhaustive Scott analysis of Peano arithmetic and linear orderings. With Montalbán the principal investigator investigated the possible Scott ranks of models of arithmetic, that is, models of the first-order theory of the natural numbers with addition and multiplication. It follows from Gödel's completeness theorem that this theory has not only
the standard model of natural numbers but many more models which extend the natural numbers by containing infinite numbers. The authors gave precise calculations of the possible Scott ranks of such models, giving a precise idea of how complicated the possible degree spectra of such structures can be and, thus, the computational complexity of such structures important to the foundations of mathematics. With Gonzalez, the principal investigator did a similar Scott analysis of models of the theory of linear orderings. Linear orderings are of interest as they are complete for Borel reducibility but not for any stronger known reducibilities. They gave a full characterization of the possible Scott sentence complexities and showed that the "incompleteness" phenomenon present in linear orderings is not reflected in their Scott analysis.
During the return phase at Technische Universität Wien, the principal investigator started several projects following up on the above highlights with local researchers and researchers across Europe. In order to attract the interest of the local scientific community they gave several talks at seminars and organized a workshop at Technische Universität Wien in the summer of 2024. For three days, 12 international researchers presented their work at this workshop titled "Computable Structure Theory and Interactions" and collaborated in Vienna.
