Frank Plumpton Ramsey.

Banach spaces and Ramsey theory, Spring 2007

Instructor: Christian Rosendal, room 304 Altgeld Hall

Course hours: Tuesday and Thursday 9:00 - 10:20 AM in Altgeld Hall 141.

Classes begin 12-Mar-07 and end 02-May-07.

This is a half semester course on the geometry of Banach spaces with a heavy emphasis on the applications of combinatorial, model-theoretical, and descriptive set theoretical methods. The goal is to introduce the fundamental notions about Schauder bases, which are the natural generalisation of vector-space bases to the infinite dimension, and then rapidly go on to prove some of the deep results of the subject.

An overview:
  • Fundamentals about bases.
  • Theorems of James.
  • Nash-Williams, Galvin - Prikry, and Rosenthal's $l_1$ theorem.
  • Spreading models and Krivine's theorem.
  • Gowers' dichotomy theorem.
  • Gowers' $c_0$ theorem.
  • Dvoretzky's theorem.

  • Notes:
  • Schauder Bases.
  • Distortion.
  • Infinite Dimensional Ramsey Theory.
  • Embeddability of $l_1$.
  • Finite Representability.
  • Unconditional basic sequences.
  • Gowers' dichotomy theorem.
  • Krivine's theorem.

  • There is no required reading, but several excellent textbooks are available.
  • F. Albiac and N. Kalton, Topics in Banach space theory.
    Graduate Texts in Mathematics, 233. Springer, New York, 2006. xii+373 pp.
  • M. Fabian, P. Habala, P. Hajek, V. Montesinos Santalucia, J. Pelant, V. Zizler, Functional analysis and infinite-dimensional geometry.
    CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC, 8. Springer-Verlag, New York, 2001. x+451 pp.
  • J. Diestel, Sequences and series in Banach spaces.
    Graduate Texts in Mathematics, 92. Springer-Verlag, New York, 1984. xii+261 pp.
  • S. Argyros and S. Todorcevic, Ramsey methods in analysis.
    Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2005. viii+257 pp.
  • D. Li and H. Queffelec, Introduction à l'etude des espaces de Banach. Analyse et probabilites.
    Cours Specialises, 12. Societe Mathematique de France, Paris, 2004. xxiv+627 pp.

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