Strategical Ramsey theory was developed in the nineties by Gowers in the
setting of Banach spaces; in this setting where the natural pigeonhole
principle does not always hold, this theory is an alternative to standard
infinite-dimensional Ramsey theory.
In this talk, I will present the formalism of Gowers spaces, an abstract
formalism unifying both strategical and standard infinite-dimensional
Ramsey theory. In this formalism, we can prove an abstract Ramsey theorem
implying both Gowers' Ramsey-type theorem in Banach spaces, and more
standard Ramsey results like Galvin-Prikry's theorem. I will also present
a result unifying infinite-dimensional Ramsey theory and determinacy.
I will then introduce a new family of Gowers spaces that arose from a
recent work in progress with Wilson Cuellar-Carrera and Valentin Ferenczi.
These examples from functional analysis are based on local properties of
subspaces of Banach spaces. We hope that examples of the same kind could
be found in other areas of mathematics.