Toronto Set Theory

Event Information Gowers spaces: unifying standard and strategical Ramsey theory
13:30 on Friday November 16, 2018
15:00 on Friday November 16, 2018
FI210, Fields Institute, 222 College St.
Noé de Rancourt
http://l'université Paris Diderot - Paris 7
Institut Mathématique de Jussieu, Paris

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.