{
  "id": "tingleys-problem",
  "title": "Tingley's problem",
  "description": "OpenAI's solution (2026) of Tingley's problem: every surjective isometry between the unit spheres of real normed spaces extends to a surjective real-linear isometry of the whole spaces. The course proves the Mazur–Ulam theorem, makes a hypothetical positive defect attained in an ultrapower, finds aligned extremal chords as a fixed point of a return map with Darbo's theorem (proved from Schauder's and Brouwer's), and rules them out by support, secant and convexity estimates.",
  "units": [
    {
      "id": "supports-radial-contact-and-the-mazur-ulam-theorem",
      "title": "Supports, radial contact and the Mazur–Ulam theorem",
      "reader": "supports-radial-contact-and-the-mazur-ulam-theorem.html",
      "source": "src/supports-radial-contact-and-the-mazur-ulam-theorem.md",
      "source_sha256": "57200acc02dfe91b9a05b7e1b8f37a0711b12a74e6559f790ba407f5138e8875",
      "status": "author_self_checked"
    },
    {
      "id": "ultrapowers-and-darbos-fixed-point-theorem",
      "title": "Ultrapowers and Darbo's fixed-point theorem",
      "reader": "ultrapowers-and-darbos-fixed-point-theorem.html",
      "source": "src/ultrapowers-and-darbos-fixed-point-theorem.md",
      "source_sha256": "c23dfde3a285eead7fb34f6484977dbbeace4296cbd52d5ce4dacd0d3c63b9c3",
      "status": "author_self_checked"
    },
    {
      "id": "aligned-chords",
      "title": "Aligned chords",
      "reader": "aligned-chords.html",
      "source": "src/aligned-chords.md",
      "source_sha256": "d0a5713a7d4cee5ed5017eafd8c4ef9140489f836bacf383759e38d5f916ed55",
      "status": "author_self_checked"
    },
    {
      "id": "tingleys-problem",
      "title": "Tingley's problem",
      "reader": "tingleys-problem.html",
      "source": "src/tingleys-problem.md",
      "source_sha256": "0f02dbd1c7c3fc78a44c8340bb8392fe8cff93497dab60aa0edd4aaba8a45203",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "foundations-of-von-neumann-algebras lesson hahn-banach-baire-and-the-basic-theorems-on-banach-spaces (Theorem 1.1, Zorn's lemma; Corollary 2.3, Hahn-Banach for normed spaces); lesson weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian (Theorem 3.1, only in an exercise)",
    "index-theory-of-elliptic-operators lesson lebesgue-s-covering-theorem-and-the-dimension-of-cubes (Corollary 1.2, Brouwer's theorem for compact convex subsets of R^n, from core course D60)",
    "real analysis: intermediate value theorem, convex functions of one variable, compactness in R^n"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}