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.