This book focuses on applications of martingales to the geometry of Banach spaces, and is accessible to graduate students.

Gilles Pisier is Emeritus Professor at the University of Paris VI, where he worked from 1981 to 2010. He is also a Distinguished Professor and holder of the Owen Chair in Mathematics at Texas A&M University. His international prizes include the Salem Prize in harmonic analysis (1979), the Ostrowski Prize (1997), and the Stefan Banach Medal (2001). He is a member of the Paris Académie des Sciences, a Foreign member of the Polish and Indian Academies of Science, and a Fellow of both the IMS and the AMS. He is also the author of several books, notably The Volume of Convex Bodies and Banach Space Geometry (Cambridge, 1989) and Introduction to Operator Space Theory (Cambridge, 2003).

Introduction; Description of the contents; 1. Banach space valued martingales; 2. Radon Nikodým property; 3. Harmonic functions and RNP; 4. Analytic functions and ARNP; 5. The UMD property for Banach spaces; 6. Hilbert transform and UMD Banach spaces; 7. Banach space valued H1 and BMO; 8. Interpolation methods; 9. The strong p-variation of martingales; 10. Uniformly convex of martingales; 11. Super-reflexivity; 12. Interpolation and strong p-variation; 13. Martingales and metric spaces; 14. Martingales in non-commutative LP *.