{
  "id": "torsion-free-group-algebra-with-zero-divisors",
  "title": "A torsion-free group algebra with zero divisors",
  "description": "There is a finitely presented torsion-free group G, with a finite two-dimensional classifying space, whose group algebra over the field with two elements has zero divisors: Kaplansky's zero-divisor conjecture fails. Two random immersed labelled graphs with prescribed letter sets of odd intersections, built from the projective plane over the field with 128 elements, make every contribution to a product cancel in pairs; a word-counting estimate, planar separation and cone pictures show that the group of the graphs is torsion-free and that the factors are nonzero. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "group-algebras-and-zero-divisors",
      "title": "Group algebras and zero divisors",
      "reader": "group-algebras-and-zero-divisors.html",
      "source": "src/group-algebras-and-zero-divisors.md",
      "source_sha256": "13c8a43283e0e2492c333c29f2b634cf4e7895df7b39d0f09a332323f3839690",
      "status": "author_self_checked"
    },
    {
      "id": "projective-planes-and-prescribed-types",
      "title": "Projective planes and prescribed types",
      "reader": "projective-planes-and-prescribed-types.html",
      "source": "src/projective-planes-and-prescribed-types.md",
      "source_sha256": "9826976fcc3499330fa5667314af57b490c4fd5d0391d1b2c9dcce01a7a66202",
      "status": "author_self_checked"
    },
    {
      "id": "random-matchings-with-large-girth",
      "title": "Random matchings with large girth",
      "reader": "random-matchings-with-large-girth.html",
      "source": "src/random-matchings-with-large-girth.md",
      "source_sha256": "1e220cd7ad18730535a59e4e03ef08a8484ff21c992d807297d5e74384058242",
      "status": "author_self_checked"
    },
    {
      "id": "image-graphs-and-multiplicity-stages",
      "title": "Image graphs and multiplicity stages",
      "reader": "image-graphs-and-multiplicity-stages.html",
      "source": "src/image-graphs-and-multiplicity-stages.md",
      "source_sha256": "e0a535aa1d844b2e7cfc62d58a4bdb6e057f493c8fe44404384652e14d05391e",
      "status": "author_self_checked"
    },
    {
      "id": "aligned-blocks-and-word-counts",
      "title": "Aligned blocks and word counts",
      "reader": "aligned-blocks-and-word-counts.html",
      "source": "src/aligned-blocks-and-word-counts.md",
      "source_sha256": "e14f74bdf95f950e417e5e97031336f9b441c65cef1a09f10a6c70f54882fa5e",
      "status": "author_self_checked"
    },
    {
      "id": "the-bounded-pattern-estimate",
      "title": "The bounded-pattern estimate",
      "reader": "the-bounded-pattern-estimate.html",
      "source": "src/the-bounded-pattern-estimate.md",
      "source_sha256": "503c5a95bc65d2515ce5f980508348490d1a472f079989cb30f8641640b6d189",
      "status": "author_self_checked"
    },
    {
      "id": "spherical-arrangements-and-planar-intervals",
      "title": "Spherical arrangements and planar intervals",
      "reader": "spherical-arrangements-and-planar-intervals.html",
      "source": "src/spherical-arrangements-and-planar-intervals.md",
      "source_sha256": "f30bdae736efcc3538c92a49eaaf5be351b91fbac9a44e78a946212c682fea5b",
      "status": "author_self_checked"
    },
    {
      "id": "planar-separation-and-the-absence-of-arrangements",
      "title": "Planar separation and the absence of arrangements",
      "reader": "planar-separation-and-the-absence-of-arrangements.html",
      "source": "src/planar-separation-and-the-absence-of-arrangements.md",
      "source_sha256": "160362c23ddc8309fd7b4814bf486518fc637eed6b98763dd52d259d83ff1b33",
      "status": "author_self_checked"
    },
    {
      "id": "cone-pictures-and-asphericity",
      "title": "Cone pictures and asphericity",
      "reader": "cone-pictures-and-asphericity.html",
      "source": "src/cone-pictures-and-asphericity.md",
      "source_sha256": "40e1456efab21f6c5b504ab16d0c5eb6920a0d0ab0299e403c9af4f18cabb820",
      "status": "author_self_checked"
    },
    {
      "id": "torsion-freeness-and-the-zero-divisors",
      "title": "Torsion-freeness and the zero divisors",
      "reader": "torsion-freeness-and-the-zero-divisors.html",
      "source": "src/torsion-freeness-and-the-zero-divisors.md",
      "source_sha256": "eb5029b42ba4e19a25776eb253883b146928eaad1737e428f0d62a89cbf773cc",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Abstract Algebra II (Judson, Abstract Algebra: Theory and Applications): the field with 128 elements (Theorem 22.1.6), coordinates and dimension in vector spaces (Propositions 20.3.1 and 20.3.6)",
    "core course Algebraic Topology (Fomberg's notes): homotopy invariance of singular homology (Theorem 1.13, Corollary 1.14); homotopy lifting for coverings (Roberts's lectures, Lecture 9, Corollary 9.4)",
    "programme lesson Explicit CW models and the first Hurewicz map (course CG-S6), Theorem 4.1: the Hurewicz theorem",
    "programme course Cell complexes, fundamental groups and covering spaces (cell-complexes-and-covering-spaces): Cell complexes and homotopy equivalences, Propositions 5.1 and 5.2; Fundamental groups of graphs and two-complexes, Corollary 2.4 and Theorem 3.3; Universal covers and cellular chains, Theorem 1.1, Proposition 2.2, Corollary 2.3, Section 5 and Proposition 5.2",
    "programme lesson Homotopy fibres and the Serre spectral sequence (course DG-CHAR), Section B.4: Whitehead's theorem for CW complexes",
    "finite probability: expectations, Markov's inequality and the union bound"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}