{
  "id": "nonsingular-systems-over-groups",
  "title": "Nonsingular systems of equations over groups",
  "description": "A system of equations over a group G has coefficients in G and unknowns x_1, ..., x_n; it is nonsingular if the matrix of exponent sums of the unknowns has full row rank over Q. Howie conjectured that every nonsingular system has a solution in some group containing G; OpenAI proved this in 2026 for every group, which includes the Kervaire-Laudenbach conjecture. The proof draws a relation that kills an element of G as a planar surface of disks and bands and measures its boundary words with the spectral phase of unitary operators over the group von Neumann algebra. The new ingredient is a theorem on simultaneous fixed spaces of block unitary matrices in several unitary variables, proved with the rational cohomology of unitary groups. The course is an exposition of OpenAI's proof (2026) and builds on the course on the Kervaire conjecture.",
  "units": [
    {
      "id": "systems-of-equations-over-groups",
      "title": "Systems of equations over groups",
      "reader": "systems-of-equations-over-groups.html",
      "source": "src/systems-of-equations-over-groups.md",
      "source_sha256": "1c36f75c87e52d5949eda56e6e1e6b4db0b28e43affc1b5c8a7488aae282f22a",
      "status": "author_self_checked"
    },
    {
      "id": "rational-cohomology-of-unitary-groups",
      "title": "Rational cohomology of unitary groups",
      "reader": "rational-cohomology-of-unitary-groups.html",
      "source": "src/rational-cohomology-of-unitary-groups.md",
      "source_sha256": "a68443c939603aa0d6ff306b1c4fc9cced1c8e9a9c5a1f7579ad4c0e77802a2b",
      "status": "author_self_checked"
    },
    {
      "id": "simultaneous-fixed-spaces",
      "title": "Simultaneous fixed spaces",
      "reader": "simultaneous-fixed-spaces.html",
      "source": "src/simultaneous-fixed-spaces.md",
      "source_sha256": "ca601961f0c32059a3b04640d2e9e65110dfd57f251547cff880d95a2965c3f3",
      "status": "author_self_checked"
    },
    {
      "id": "the-planar-boundary-obstruction",
      "title": "The planar boundary obstruction",
      "reader": "the-planar-boundary-obstruction.html",
      "source": "src/the-planar-boundary-obstruction.md",
      "source_sha256": "5f555a9e9d79397b9decdb8320c9fb18fe42e2372ec046eeae58d444412f76f3",
      "status": "author_self_checked"
    },
    {
      "id": "from-a-kernel-relation-to-a-solution",
      "title": "From a kernel relation to a solution",
      "reader": "from-a-kernel-relation-to-a-solution.html",
      "source": "src/from-a-kernel-relation-to-a-solution.md",
      "source_sha256": "a4173c55411187625525e3f5c2c0e00844a3fd80ac3c3e8e628e6081a1f101b2",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "programme course The Kervaire conjecture (the-kervaire-conjecture): lessons The regular trace and spectral phase (Lemmas 1.3, 2.1, 3.1, 4.1), A unitary with a large fixed space (Lemmas 1.1, 2.1, 5.1), Disks, bands and boundary words ((SURF), Lemmas 3.1, 4.2, 5.1, Theorem 2.1), The planar boundary theorem ((0.1), Lemma 3.1), From a relation to a planar surface (Lemmas 1.1, 3.1, 3.2, the extension of loops over disks)",
    "programme course Poincaré duality on manifolds: lesson Poincaré duality, Corollary 2.1",
    "programme course torsion-free-group-algebra-with-zero-divisors: the named statement (SURF) and the statement (PI1)",
    "programme course Cell complexes, fundamental groups and covering spaces (cell-complexes-and-covering-spaces): lesson Fundamental groups of graphs and two-complexes, Corollary 2.4 and Theorem 3.3, which prove the statement (PI1)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}