# Applications of infinitary combinatorics in Banach Spaces of the forms $C(K)$, $C(K\times L)$, $C(K,X)$ and related structures
 Grant number: 13/20703-6 Support type: Scholarships abroad - Research Internship - Post-doctor Effective date (Start): February 01, 2014 Effective date (End): January 31, 2015 Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis Principal Investigator: Eloi Medina Galego Grantee: Leandro Candido Batista Supervisor abroad: Piotr Koszmider Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil Local de pesquisa : Polish Academy of Sciences (PAN), Poland Associated to the scholarship: 12/15957-6 - On Banach Spaces $C_0 (K, X)$ and the topology of $k$, BP.PD Abstract The scientific goal of the project is to contribute in several directions to the understanding of the Banach spaces of continuous functions and related structures. Although not foreseen in the original postdoctoral project, this internship abroad comply with it and is an excellent opportunity to further develop the research initially proposed. This project is intended to be supervised by Prof. Piotr Koszmider and conducted from 01 February 2014 to 31 January 2015 at the Mathematical Institute of the Polish Academy of Sciences in Warsaw, Poland, where the supervisor is employed as an extraordinary professor. The project will start with a deep investigation on the interaction between the classical theory of Banach spaces $C(K)$ and modern methods of the infinitary combinatorics. We are mainly interested to continue the work of the candidate in his postdoctoral adding this entirely new approach. With this aim in mind, we are lead to consider the Banach spaces $C(K,C(L))$, where $K$ and $L$ are infinite compact Hausdorff spaces. Since $C(K,C(L))$ is isometric isomorphic to $C(K\times L)$, these spaces seem to offer a prominent enviroment for this research. It is important to note that the geometry of the Banach space $C(K \times K)$ may differ drastically from the geometry of $C(K)$ when $K$ is an infinite compact Haursdorff space, for example, if $\beta \mathbb{N}$ denotes the Stone-\v Cech compactification of the natural numbers, it is well known that $C(\beta \mathbb{N})$ contains no complemented copy of the Banach space $c_0$ of sequences that converge to zero, while $C(\beta \mathbb{N} \times \beta \mathbb{N})$ does contain such a copy. This example shows that the study of the spaces $C(K,C(L))$ shall employ and inspire the development of new tools for investigating the Banach spaces of continuous functions.Gradually, during the project, we will search for applications of these new approaches for $C(K, X)$ spaces, when $X$ is a Banach space containing no copy of $c_0$. There is also the expectation that Prof. Koszmider's methods can be succesfully applied to Infinite-dimensional holomorphy, a topic where the candidate received his Master degree and has some experience. Arguably, the Banach spaces $C(K)$ can be studied by using principles of the infinitary combinatorics, since one can define a Boolean algebra for which, according to Stone's representation theorem, it is possible to obtain a compact space $K$, then consider the Banach space $C(K)$ and study how analytic properties of this space are influenced by combinatoric properties of the Boolean algebra. When working in this field, it is commom to find some situations where the problems are solved by using some extra set-theoretic principles which cannot be established on the basis of the standard axiomatic system (ZFC). On the other hand, the application of another set-theoretic assuptions may solve the problem in the opposite direction, establishing it as undecidable on the basis of ZFC. The scientific background of the supervisor allows him to handle such situations and advise the candidate in this field.The main topics of this project are motivated by recent directions of the international research, sometimes marked by contributions of the supervisor of this project, and the research being developed by the candidate. They include:(1) On Banach spaces $C(K)$ with few operators and the geometry of $C(K\times K)$, (2) The Banach space $C(K)$ with $K$ scattered or with the Radon-Nikod\'ym property, (3) Biorthogonal systems and semibiorthogonal sequences, (4) Properties of compact space $K$ detected by Banach spaces $C(K,X)$, (5) Infinite-dimensional holomorphy. (AU)