Set-theoretic combinatorics in Banach and measure spaces

With its broad applications in such sciences as physics, engineering, biology or medicine, analysis constitutes one of the main branches of modern mathematics. It is intensively studied by many mathematicians, however it often happens that some hypotheses of analysis cannot be proved as true or refuted as false with only use of analytical techniques. The problem usually lies in the assumed collection of axioms of mathematicsi.e. sentences taken as true without a proof and from which every theorem of mathematics is derivedconcerning the notion of a set. Sets are considered as the most minimal objects in mathematics: all other objects such as numbers, functions or spaces are in fact built out of sets. However, despite the minimal character of sets, even a slightest change in the collection of the axioms may have a great impact on whole mathematics, implying that some mathematical objects will start or will stop to exist, some relations between objects will start or will stop to hold, or various objects will start or will stop to have certain properties. The branch of mathematics studying properties of axioms of sets and their impact on the rest of mathematics is called set theory. In this project we are interested in investigating what the impact of set theory on the existence and structure of analytical spaces having certain properties is. We especially ask from the set-theoretic point of view about spaces having various properties concerning convergence of infinite sequences of their elements, spaces in which we are able to conduct certain measurements, or spaces containing other spaces. The method of forcing will constitute our main research tool. This is a set-theoretic technique which allows us to create new mathematical worlds (or universes) in which certain mathematical objects or spaces exist. We will also use so-called cardinal characteristics of the continuum, that is, objects that are a bit more complicated than sets such as certain families of infinite sequences of natural numbers or families of special infinite subsets of the set of real numbers. Those techniques have been deeply studied in set theory and now are well-understood. One of the main innovative aspects of the project is to use them in the investigation of the above-mentioned problems concerning spaces originating in analysis. This application will reveal the combinatorial structure of certain spaces used in analysis, and hence improve our understanding of them, but will also allow to prove that various important questions concerning them are undecidable, that is, depending on the assumed set of axioms they can be either proved or refuted.

With its broad applications in such branches of science as physics, engineering, biology or medicine, analysis constitutes one of the main research areas of modern mathematics. In this project we were interested in investigating various properties of so-called Banach spaces, which are one of the central objects and tools of analysis. Those spaces can be considered as a natural infinite-dimensional generalization of typical finite-dimensional Euclidean spaces such as the standard 2-dimensional plane or 3-dimensional space. During the project we obtained a series of results describing the structure, geometry, dimension, and combinatorial features of those spaces, as well as their possible transformations and deformations through various continuous operations, e.g. those which preserve the distances between points or the additive vector structure of the spaces. To obtain those results, next to applying standard methods characteristic for analysis and Banach space theory, we also often exploited tools originating from such branches of mathematics as logic and topology-such an approach allowed us to investigate the spaces also from the more fundamental or logical point of view, often unraveling hidden properties of the spaces which heavily depended on the assumed system of axioms of mathematics.