{
  "id": "metric-markov-cotype-two-of-l1",
  "title": "Metric Markov cotype two of ℓ₁",
  "description": "Metric Markov cotype two: moving points on the states of a reversible Markov chain so that neighbouring states carry nearby points, at a cost controlled by the variation of the old points over a time interval. The theorem: the sequence space ℓ₁ has this property, with an explicit constant. Its proof: cuts of finitely many points, a cubic map of a cube that is flat at the vertices, a fourth-power potential, and a walk that stops at a geometric random time; the new points are means of medians of three. Also: Hilbert space, the spaces L₁(μ), complex scalars, and the extension of Lipschitz maps from subsets of Hilbert space into ℓ₁ (from one named hypothesis, the extension theorem of Mendel and Naor). The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "metric-markov-cotype-two",
      "title": "Metric Markov cotype two",
      "reader": "metric-markov-cotype-two.html",
      "source": "src/metric-markov-cotype-two.md",
      "source_sha256": "06a69debf57c101831ace8ecbabb925e391c5bbc2d8549fe186bbfb5d9e4f8c8",
      "status": "author_self_checked"
    },
    {
      "id": "cuts-the-cubic-map-and-a-quartic-potential",
      "title": "Cuts, the cubic map and a quartic potential",
      "reader": "cuts-the-cubic-map-and-a-quartic-potential.html",
      "source": "src/cuts-the-cubic-map-and-a-quartic-potential.md",
      "source_sha256": "c5039b2356b35775843c48e05a31e63ff924ca7c8202594c6a0847548686a4e1",
      "status": "author_self_checked"
    },
    {
      "id": "the-stopped-walk-and-the-cotype-inequality",
      "title": "The stopped walk and the cotype inequality",
      "reader": "the-stopped-walk-and-the-cotype-inequality.html",
      "source": "src/the-stopped-walk-and-the-cotype-inequality.md",
      "source_sha256": "f7948a3a9e41c7d959f4c1cfa874720729e19e0bf7917b2bd5865e0b0b9e2963",
      "status": "author_self_checked"
    },
    {
      "id": "complex-scalars-measure-spaces-and-lipschitz-extension",
      "title": "Complex scalars, measure spaces and Lipschitz extension",
      "reader": "complex-scalars-measure-spaces-and-lipschitz-extension.html",
      "source": "src/complex-scalars-measure-spaces-and-lipschitz-extension.md",
      "source_sha256": "2121fa60ee577e86de626c34bb153f67220560203ece459e3e1cf1c0ed5cff5c",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "finite sums, the Cauchy-Schwarz inequality, geometric series",
    "core course Linear Algebra (B40): the spectral theorem for symmetric matrices (first lesson, Section 3; fourth lesson, Section 3)",
    "core course Measure and Integration (D10): integrable functions (fourth lesson, Section 2 only)",
    "core course Functional Analysis (D20): normed spaces and bounded linear functionals (fourth lesson, Section 4 only)",
    "the lessons prove every other fact they use, except the named hypothesis (MN) of the fourth lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}