Markov type and superreflexivity
Markov type of normed spaces: how far a stationary reversible Markov chain can travel in t steps. Uniformly convex norms, superreflexivity, finite representability. The theorem: a real Banach space with Markov type p > 1 is superreflexive. Its proof: a norm on sequences that remembers the order of the entries, random increasing maps and a reversible chain of them, four homeomorphisms of the line and words in a free group, and a reversible walk that moves at linear speed. One named hypothesis, the theorem of James and Enflo; everything else is proved. The course is an exposition of OpenAI's proof (2026).
Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Markov type and the reduction to a finite walk
- A spreading norm with contractive merging
- Charts and reversible transport
- Two levels from reduced words
- The finite walk
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Alignment of lists: patterns, minimax and Ramsey's theorem (Large independent sets in three-colorable graphs)
- Normed spaces, the bidual and the Hahn–Banach theorem (core course Functional Analysis); finite probability.