{
  "id": "markov-type-and-superreflexivity",
  "title": "Markov type and superreflexivity",
  "description": "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).",
  "units": [
    {
      "id": "markov-type-and-the-reduction-to-a-finite-walk",
      "title": "Markov type and the reduction to a finite walk",
      "reader": "markov-type-and-the-reduction-to-a-finite-walk.html",
      "source": "src/markov-type-and-the-reduction-to-a-finite-walk.md",
      "source_sha256": "8390833b4ca084139ae6d0fe307bc5cf9ec148def30197ce74df0fe8040fac42",
      "status": "author_self_checked"
    },
    {
      "id": "a-spreading-norm-with-contractive-merging",
      "title": "A spreading norm with contractive merging",
      "reader": "a-spreading-norm-with-contractive-merging.html",
      "source": "src/a-spreading-norm-with-contractive-merging.md",
      "source_sha256": "6eeff7967727640c4635a397d5a2741dd85c2cde353acf119c19849a22824b6b",
      "status": "author_self_checked"
    },
    {
      "id": "charts-and-reversible-transport",
      "title": "Charts and reversible transport",
      "reader": "charts-and-reversible-transport.html",
      "source": "src/charts-and-reversible-transport.md",
      "source_sha256": "a6a97deb0a2e76fe1c9c7e77c78f2729670b0d5e3050183a4ada67a984386bbb",
      "status": "author_self_checked"
    },
    {
      "id": "two-levels-from-reduced-words",
      "title": "Two levels from reduced words",
      "reader": "two-levels-from-reduced-words.html",
      "source": "src/two-levels-from-reduced-words.md",
      "source_sha256": "c5a72eeb190b6f94ac2da130cc51dac2240955077c9f8b0c3d6f9d051ffc31cf",
      "status": "author_self_checked"
    },
    {
      "id": "the-finite-walk",
      "title": "The finite walk",
      "reader": "the-finite-walk.html",
      "source": "src/the-finite-walk.md",
      "source_sha256": "eb6d514937c58170c6792ffafe37f43481b60a347071021d4533771bb7014dcb",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Functional Analysis (D20): normed spaces, the dual and the bidual, the Hahn-Banach theorem",
    "finite probability: conditional means, Markov's inequality, finite Markov chains (defined in the first lesson)",
    "Ramsey's theorem, the minimax theorem and a separation lemma, proved in the course Large independent sets in three-colorable graphs (lesson Alignment of lists: patterns, minimax and Ramsey's theorem)",
    "the lessons prove every other fact they use, except the named hypothesis (JE) of the first lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}