Analytic P-ideals, Banach spaces, and measure algebras

The project is focused on old as well as recently discovered interactions between classical mathematical objects of different fields such as combinatorics, set theory, algebra, geometry, and topology. Let us mention a couple of examples. How complicated can a finite set of natural numbers be? To see a less obvious example, consider the following property of such a set: It has at most as many elements as its smallest element. Although, the idea may feel ad hoc, the collection of all these sets is called the Schreier family and it played a fundamental role in the history of infinite dimensional geometry. From the many set-theoretic structures we are going to investigate in this context, perhaps ideals are the most important. They are collections of sets of natural numbers, just like the Schreier family but now we consider infinite sets as well, which are thin in some precise, analytic sense. At last but not least, we shall mention that the collection of all subsets of the plane which have area (most of them do not) form an algebraic structure but at the same time also played a crucial role in finding witnesses to Gödels first incompleteness theorem, that is, mathematical questions which cannot ever be answered. The interplay between these notions and structures has been discovered and studied for a long time, mostly because one can be constructed from another. Moreover, when doing so sometimes we obtain complete circles of many of these objects. Apart from the natural interest in such interactions be tween numerous fields of mathematics, these constructions provide us with new approaches to well-studied classical notions and hence with new tools when discussing these structures. This leads to new characterizations of sometimes basic properties; and in some specific cases, this approach even shed light on new possible ways when attacking long-standing open problems. The project is devoted to further develop and apply some of these old and new bridges between these fields, and hopefully to serve as a systematic foundation of many further research in this beautiful multidisciplinary area.

The project was focused on certain interactions of functional analysis and set theory, two classical fields in mathematics. We shall distinguish two areas of our contributions: (1) One of the main objects we studied was combinatorial Banach spaces. These seemingly simple spaces, formally belonging to pure functional analysis, show more and more surprising potential to connect seemingly independent fields. We further developed these interactions between geometry, topology, set theory, and added graph theory to the list. Among other results, our approach led us to (i) a plethora of new examples of combinatorial spaces witnessing the rich variety of them, (ii) an elegant solution of a 50-year old problem concerning universal Banach spaces, (iii) a completely unexpected characterization of the duality phenomenon via so-called perfect graphs, and (iv) many independence results on the set-theoretic end of our investigations. These developments and the work of other researchers have turned the study of these spaces a rapidly growing field sitting in the intersection of functional analysis, set theory, topology, and combinatorics. (2) Another object studied in the project was Banach spaces of continuous functions, constituting one of the most important classes of infinite-dimensional spaces investigated in functional analysis. Our main results concerned their geometrical and topological structure, in particular, we studied how and when it is possible to embed one such space into another while preserving the natural notion of distance between functions, and how one can decompose such space into smaller pieces. We provided several criteria for the existence of such embeddings between spaces of continuous functions which are sufficiently small, as well as, we showed that in many cases they can be represented as sums of smaller spaces of continuous functions. Our results not only provided new insight into this class of Banach spaces, but also extended and supplemented the classical theory.