Spiros A. Argyros Boeken

This book features two sets of notes from the Advanced Course on R-sey Methods in Analysis held at the Centre de Recerca Matemàtica in January 2004, part of a research program on Set Theory and its Applications. The notes aim to assist young mathematicians in exploring a dynamic research area at the intersection of analysis and combinatorics. Key recent advances discussed include the distortion problem for Hilbert space, the unconditional basic sequence problem for Banach spaces, and the Banach homogeneous space problem. The primary objective is to unveil the general principles and methods that could facilitate future developments. The first set of notes focuses on a method for constructing norms with specific properties, essential for understanding the infinite-dimensional geometry of Banach spaces. The second set delves into Ramsey-theoretic methods that illuminate the rough structure inherent in this geometry. Acknowledgments are extended to Joan Ba- ria, the course coordinator, and Manuel Castellet, the CRM director, for providing this challenging yet rewarding opportunity. Part A addresses saturated and conditional structures in Banach spaces.