{
  "id": "taming-implies-compatibility",
  "title": "Taming implies compatibility",
  "description": "Donaldson asked whether, on a closed four-manifold, an almost complex structure tamed by a symplectic form admits a compatible symplectic form. The answer is yes. If not, separation gives a positive current that annihilates closed invariant forms, and a closed lift adds an anti-invariant correction so that the sum annihilates all closed forms. Resolvent regularization shows that the correction is square integrable and that nearby complex lines of the current align; at points of positive density the current is planar, and its restriction there is closed. A final pairing removes the correction, and the current then annihilates the taming form, which is impossible. Proved in the programme from one named hypothesis, the short-time asymptotics of the heat kernel of the Hodge Laplacian. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "taming-compatibility-and-a-separating-current",
      "title": "Taming, compatibility and a separating current",
      "reader": "taming-compatibility-and-a-separating-current.html",
      "source": "src/taming-compatibility-and-a-separating-current.md",
      "source_sha256": "e9b65cd8da1cb64cadeec2c4e34f8e2a48f4014df6ea60d89ded745f4062a0d7",
      "status": "author_self_checked"
    },
    {
      "id": "quadratic-mass-growth",
      "title": "Quadratic mass growth",
      "reader": "quadratic-mass-growth.html",
      "source": "src/quadratic-mass-growth.md",
      "source_sha256": "740a6332d4fdf0f465d5837269ebd7cb5ead912983b435222beee3e2f0963584",
      "status": "author_self_checked"
    },
    {
      "id": "regularization-and-angular-control",
      "title": "Regularization and angular control",
      "reader": "regularization-and-angular-control.html",
      "source": "src/regularization-and-angular-control.md",
      "source_sha256": "2a6000fdea038e63e3f457e2172612c850467bbd7e6da33e89fc13e294620ded",
      "status": "author_self_checked"
    },
    {
      "id": "densities-and-planar-tangent-measures",
      "title": "Densities and planar tangent measures",
      "reader": "densities-and-planar-tangent-measures.html",
      "source": "src/densities-and-planar-tangent-measures.md",
      "source_sha256": "f524829a620da8e4d732e7c17f5b55a71350440aa949e9f79849ec6a7b3b8482",
      "status": "author_self_checked"
    },
    {
      "id": "closedness-of-the-density-part-and-the-proof",
      "title": "Closedness of the density part and the proof",
      "reader": "closedness-of-the-density-part-and-the-proof.html",
      "source": "src/closedness-of-the-density-part-and-the-proof.md",
      "source_sha256": "57ef3ef42822eafa725a259019e21f789e66ce160590217d018b4dc2f2580ef7",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course Elliptic boundary reduction, lessons Symbols, finite defects, and the index on a closed manifold; Detecting regularity without choosing coordinates; Symbols, operators and Sobolev scales: bundle Sobolev spaces, the elliptic alternative, operators of order zero and their composition",
    "course Bundles, curvature and holonomy, lessons Geodesics, normal coordinates and curvature; Riemannian connections and convex neighbourhoods; Jacobi fields, conjugate points and the Morse index theorem; Connections and parallel transport",
    "course Distributions, kernels and analytic singularities, lesson Order, positivity and distributional limits: positive linear forms are measures",
    "course Measurable fields and direct integrals, lesson Besicovitch's covering theorem and the differentiation of Radon measures",
    "course Foundations of von Neumann algebras, lessons Hahn-Banach, Baire and the basic theorems on Banach spaces; Hilbert spaces and compact operators; Weak topologies",
    "core course Measure and Integration: the Radon-Nikodym theorem, image and product measures",
    "named hypothesis (HK) on the heat kernel of the Hodge Laplacian, stated in the third lesson; not proved in the programme"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}