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.