{
  "id": "semidefinite-lifts-of-the-perfect-matching-polytope",
  "title": "Semidefinite lifts of the perfect matching polytope",
  "description": "The shifted matching matrices, with entries |M ∩ δ(U)| − 1 + ρ for odd sets U and perfect matchings M of the complete graph K_n, have positive semidefinite rank 2^{Ω(n)} for every fixed 0 < ρ < 1; consequently every exact semidefinite lift of the perfect matching polytope has exponential size. Averaging row factors over restrictions of cuts to cells, an orthogonal splitting inequality and a Fourier smoothing theorem turn a small factorization into an approximation of a parity function on an expanding graph by squares of low degree, which a functional of Grigoriev type rules out. The course is an exposition of OpenAI's proof (2026), with a further smoothing theorem for averages over perfect matchings.",
  "units": [
    {
      "id": "psd-factorizations-and-semidefinite-lifts",
      "title": "PSD factorizations and semidefinite lifts",
      "reader": "psd-factorizations-and-semidefinite-lifts.html",
      "source": "src/psd-factorizations-and-semidefinite-lifts.md",
      "source_sha256": "d994dc578f23232f8d55a54d4a167b2fa1efe51ca03a264451ec23166a696192",
      "status": "author_self_checked"
    },
    {
      "id": "averaging-within-a-cell",
      "title": "Averaging within a cell",
      "reader": "averaging-within-a-cell.html",
      "source": "src/averaging-within-a-cell.md",
      "source_sha256": "26862b4a3aec54f9cce1e3f6fa9dce4b9cb108f5278f419245328c492a071e7a",
      "status": "author_self_checked"
    },
    {
      "id": "splitting-positive-semidefinite-matrices",
      "title": "Splitting positive semidefinite matrices",
      "reader": "splitting-positive-semidefinite-matrices.html",
      "source": "src/splitting-positive-semidefinite-matrices.md",
      "source_sha256": "0f1ca270c695c0d4a053cfcc75f99e11c25f92b0ec20e85e2943465f6ed0ed72",
      "status": "author_self_checked"
    },
    {
      "id": "fourier-smoothing-of-matrix-squares",
      "title": "Fourier smoothing of matrix squares",
      "reader": "fourier-smoothing-of-matrix-squares.html",
      "source": "src/fourier-smoothing-of-matrix-squares.md",
      "source_sha256": "2a24eba9eb2a6dfdeb825cca056ee9cf0840cb3067a61733d6549b4ec2deb8f1",
      "status": "author_self_checked"
    },
    {
      "id": "an-expanding-parity-system",
      "title": "An expanding parity system",
      "reader": "an-expanding-parity-system.html",
      "source": "src/an-expanding-parity-system.md",
      "source_sha256": "8d8a19dd22462e2d9b5efc8fc4de1554126640ba47fdba3a9e82645569274419",
      "status": "author_self_checked"
    },
    {
      "id": "the-lower-bound-for-perfect-matching",
      "title": "The lower bound for perfect matching",
      "reader": "the-lower-bound-for-perfect-matching.html",
      "source": "src/the-lower-bound-for-perfect-matching.md",
      "source_sha256": "74a6df112c966f44cd7fa950b637c43860315c5e1bf6095f399b08944586f510",
      "status": "author_self_checked"
    },
    {
      "id": "uniform-smoothing-of-matching-averages",
      "title": "Uniform smoothing of matching averages",
      "reader": "uniform-smoothing-of-matching-averages.html",
      "source": "src/uniform-smoothing-of-matching-averages.md",
      "source_sha256": "f753aab389a14a156c439cf692e426ca2ec532344f61a309381978f19e6328cc",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "finite-dimensional linear algebra: the spectral theorem for real symmetric matrices",
    "basic real analysis: compactness of closed bounded sets in finite dimensions, comparison of sums with integrals",
    "course RT-FIN (Representations of finite groups), lessons The symmetric groups I: Young tableaux and Young symmetrizers (Theorem 4.3, Proposition 5.2 and the Young symmetrizer (5)) and The symmetric groups II: branching, Jucys-Murphy elements and Young's seminormal form (Theorem 4.2 and Corollary 4.3); used only in the seventh lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}