{
  "id": "stability-of-brenier-maps",
  "title": "Stability of Brenier maps",
  "description": "For the uniform measure on a convex body, quadratic optimal transport maps depend on the target in a one-third Hölder way for the Wasserstein distance, with one constant for all targets in a fixed compact set. The exponent one third is optimal, already on a cube with three-atom targets, which disproves a conjectured square-root estimate. The proof: derivatives of Laguerre cells; a uniform estimate of moment changes by mass changes, from a midpoint coupling by Knothe's rearrangement; moving the sites of finite targets with fixed masses keeps the potentials within O(W_2); a slice interpolation inequality turns this into the exponent one third. Proved in full in the programme, with no named hypothesis. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "optimal-maps-of-finite-affine-maxima",
      "title": "Optimal maps of finite affine maxima",
      "reader": "optimal-maps-of-finite-affine-maxima.html",
      "source": "src/optimal-maps-of-finite-affine-maxima.md",
      "source_sha256": "101676f05494f4eb1e94db26c8798b28509f5dddd92c393fe127a3b1e444075f",
      "status": "author_self_checked"
    },
    {
      "id": "laguerre-cells-and-prescribed-masses",
      "title": "Laguerre cells and prescribed masses",
      "reader": "laguerre-cells-and-prescribed-masses.html",
      "source": "src/laguerre-cells-and-prescribed-masses.md",
      "source_sha256": "0c73ab8cf1f64dd75598b3332c7d75e48e9ba6a1aa9ff0b39dafab0903ada710",
      "status": "author_self_checked"
    },
    {
      "id": "a-uniform-estimate-for-cell-moments",
      "title": "A uniform estimate for cell moments",
      "reader": "a-uniform-estimate-for-cell-moments.html",
      "source": "src/a-uniform-estimate-for-cell-moments.md",
      "source_sha256": "7b48cd21c86509296376a9b3280fbfb331ff303fd4afe3082ed3d9f2a1bda85c",
      "status": "author_self_checked"
    },
    {
      "id": "moving-sites-with-fixed-masses",
      "title": "Moving sites with fixed masses",
      "reader": "moving-sites-with-fixed-masses.html",
      "source": "src/moving-sites-with-fixed-masses.md",
      "source_sha256": "147a3c8a3d5e0147d98ca6f8584d67ad920840a3e2c334382eca35b361f824f5",
      "status": "author_self_checked"
    },
    {
      "id": "from-potentials-to-maps",
      "title": "From potentials to maps",
      "reader": "from-potentials-to-maps.html",
      "source": "src/from-potentials-to-maps.md",
      "source_sha256": "ee6f15b9a4a2a6f9b6562e9c7b4e571b5a23156cb53a962cc7d944cd68304ad5",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Measure and Integration: Lebesgue measure, image measures, Fubini and Tonelli, dominated convergence",
    "core course Real Analysis II (Lebl, Basic Analysis II): the implicit function theorem (Theorem 8.5.6) and the Arzela-Ascoli theorem (Theorem 11.6.9)",
    "the lessons prove every other fact they use, including the geometry of convex bodies, the Knothe rearrangement and the differentiability of convex functions"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}