Project's title: Combinatorial and topological methods in the Banach spaces
of continuous functions

Individual project - 06.2008-05.2011

The subject of the project is the application of new methods of
infinitary combinatorics and set-theoretic topology in the
topics concerning unsolved until now classical problems of Banach spaces of continuous functions.
The aim of the project is solving or getting closer to solutions
of some of these problems and also obtaining conclusions in the
general geometry of Banach spaces.

New combinatorial methods mentioned above are the continuously developing
method of forcing (proper forcing, iterated forcing, OCA, PFA etc), the method
of elementary submodels and new Ramsey type principles.
On the other hand, the topological methods concern constructions of compact
spaces, quite often using the Stone duality and the above combinatorial methods.

The classical unsolved problems concerning the Banach spaces of continuous functions
are for example, the injective space problem, the problem if every complemented subspace
of a C(K) is isomorphic to some C(K'), etc. These problems were often left unsolved
already in the sixties because of the difficulties of the combinatorial nature.

Since, the techniques mentioned above were developed in the last 30 years
in a relative isolation from the geometry of Banach spaces, the aim of the project is
to test them in the context of the above classical problems.
The recent work of the applicant shows that the outcome of of such a test could be
unexpectedly striking.