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# Regularity properties, definability and combinatorics on the real line.

## Periodic Reporting for period 1 - REGPROP (Regularity properties, definability and combinatorics on the real line.)

Okres sprawozdawczy: 2017-03-01 do 2019-02-28

This research project is in set theory, an area of abstract mathematics dealing with foundational questions such as "what is the nature of the real number continuum"', "what can be formally proved on the basis of commonly accepted axioms", and "what are the limitations of mathematical formalism".

The topic of this research project was regularity properties of sets of real numbers. To give an example: Lebesgue measure is the standard mathematical formalism to capture the intuition of "size" or "volume" of an object in space; a straightforward definition assuming that all sets can be measured in volume, however, leads to paradoxes (such as the famous Banach-Tarski paradox). As a consequence, we have to accept that not all sets are Lebesgue measurable: some sets are and others are not. We call Lebesgue measurability a regularity property since it separates the regular from the irregular sets.

Usually, sets with simple definitions are regular while sets with complicated definitions can be irregular. One of the goals of set theory is to understand the extent of regularity in the universe of sets and characterise exactly what the conditions for regularity and irregularity as well as the interplay between different regularity properties is.

The study of regularity properties is a well-established subfield of set theory to which this project contributed by providing an extension of the analysis to more complicated sets, providing new examples of regularity properties, and developing new techniques for their study.
This project contributed to the study of regularity properties by providing a more abstract analysis, by applying known techniques to new regularity properties, and by generalising the techniques to other fields of mathematics. We give three examples:

Abstract Analysis. The general method for linking regularity properties to the existence of generic reals (due to Ikegami) was extended to a more general setting, allowing for an analysis of the case of Amoeba forcing (a case that heretofore was not covered by the technique).

Application of known techniques. The standard techniques for analysis of regularity properties on the second level of the projective hierarchy were applied to regularity properties that they have not been applied to before.

Generalising the techniques. Recently, the study of generalised Baire spaces (analogues of the real number continuum at higher cardinals) has become a trending topic in set theory and this project contributed to the research in the field by studying the generalised version of Laver forcing, a forcing well-known from the classical case.

The results of the project were obtained in numerous collaborations with researchers from Austria, Finland, Germany, Israel, Japan, the Netherlands, and Poland in the form of research visits of project members as well as invitations to Hamburg. The results are or are going to be published in research journals and furthermore presented at a number of leading research conferences in logic and mathematics in general, e.g. the Asian Logic Conference, Logic Colloquium, and the annual meeting of the German Maths Society. In addition, members of the project gave numerous presentations in local, national, and international research seminars.
This project has made important contributions to set theory, pushing the boundaries beyond the state of the art and opening up new possibilities, particularly in regard to Generalized Baire Space and higher complexity levels.

Research in pure mathematics is fundamental research that does not have direct societal impact. Its impact is usually indirect and (very) long-term, manifested by deeper understanding of the mathematical concepts and the training of future researchers in the field. Of course, research in the foundations of mathematics is relevant and necessary since all applications of mathematics, and by extension science at large, are based on these foundations.

The project contributed to opening up new research areas; this will have an immediate effect on the research community and by that a potential long-term impact on the development of the field. By involving junior researchers and research students, it has directly contributed to the training of a new generation of researchers (as witnessed by Master's and doctoral theses written in close coordination with the project aims).