## Weekly outline

### Introduction

**This course is now over.**The lectures took place every Wednesday 13.30-15.30 and Thursday 10.30-12.30, both in-person at Palazzo Campana and online on webex.

This was a reading course, so the participating students all presented a result from the literature.### Program

The first and central part of the course focuses on those parts of Banach space theory in which infinite combinatorics is used. This entails results characterizing infinite-dimensional Banach spaces that are not reflexive, so not isomorphic to their double-dual. A general presentation is given in the first lecture.

We first focus on Ramsey-like results and their applications, starting by giving a game-theoretical proof of the Galvin-Prikry and Silver theorems, following works of Tanaka and Kastanas (second lecture). We go on with applications to Banach space theory, first Rosenthal's characterizations of Banach spaces containing $l_1$ that uses the Galvin-Prikry theorem (third-fourth lecture). We then show how to obtain the Brunel-Sucheston theorem on the existence of a spreading model as a consequence of Ramsey's theorem (fourth-fifth lecture).

We then use this to see Rosenthal's characterization of those sequences whose spreading model is isomorphic to $l_1$ (fifth-sixth lecture).

We conclude the central part of the course by seeing the characterization of those spaces containing $l_1$ or $c_0$, using the double-dual space by Odell-Rosenthal, that connects Banach spaces to the first Baire class (sixth-eighth lecture).

This first central part of the course is concluded by a short summary putting in perspective various results that have motivated and driven the results that have been seen so far (end of eighth lecture).

In a second moment, we shall follow more recent works that stem from the central part. We start with exploring the descriptive complexity of reflexive separable spaces, following Dodos's book on the matter (ninth and tenth lecture).

We then have a presentation of the existing connection between ideals, norms, measures, and spaces. We follow for this segment of the course an article by Borodulin-Nadzieja and Farkas called Analytic P-ideals and Banach spaces (eleventh and twelfth lecture).

Luca Motto Ros concludes by presenting Louveau and Rosendal's results on the descriptive complexity of linear isometric embeddability between Banach spaces.### Some notes

### Video recordings