{
  "id": "kaplansky-direct-finiteness",
  "title": "Kaplansky's direct-finiteness conjecture",
  "description": "Does ab = 1 imply ba = 1 in every group algebra? Kaplansky asked this, and the answer is yes in characteristic zero and for sofic groups. OpenAI showed in 2026 that it is no in general: over a finite field of characteristic two there are a finitely presented group G and a, b with ab = 1 and ba != 1. The group is not sofic and carries an injective cellular automaton that is not surjective, disproving Gottschalk's conjecture. The course proves Kaplansky's theorem in characteristic zero and the Gromov-Weiss theorem, then the construction: copies of F_l^2 glued along a system of affine lines of incidence girth 12, character packing, low-degree interpolation, and a line system built from holes, random slabs and the local lemma. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "directly-finite-group-algebras",
      "title": "Directly finite group algebras",
      "reader": "directly-finite-group-algebras.html",
      "source": "src/directly-finite-group-algebras.md",
      "source_sha256": "419ed564a9fa4f12da35fa47a6478f22e62d8efa5c167c7de2bd61cceb4a2b92",
      "status": "author_self_checked"
    },
    {
      "id": "sofic-groups-and-surjunctivity",
      "title": "Sofic groups and surjunctivity",
      "reader": "sofic-groups-and-surjunctivity.html",
      "source": "src/sofic-groups-and-surjunctivity.md",
      "source_sha256": "557ac327326b70a8d65fef63a4a9a068594434f898a9a1173421b0232236840b",
      "status": "author_self_checked"
    },
    {
      "id": "the-glued-group-and-planar-balance",
      "title": "The glued group and planar balance",
      "reader": "the-glued-group-and-planar-balance.html",
      "source": "src/the-glued-group-and-planar-balance.md",
      "source_sha256": "7a0804c4056f4b7a18858d7828f13d3fc260f4075c321021624208cceeecbc4f",
      "status": "author_self_checked"
    },
    {
      "id": "subgroup-averages-and-character-twists",
      "title": "Subgroup averages and character twists",
      "reader": "subgroup-averages-and-character-twists.html",
      "source": "src/subgroup-averages-and-character-twists.md",
      "source_sha256": "1e5908bbd949d656bcca62013465919b2ee92bbb659d531d4707789779ab724b",
      "status": "author_self_checked"
    },
    {
      "id": "packing-and-the-reverse-defect",
      "title": "Packing and the reverse defect",
      "reader": "packing-and-the-reverse-defect.html",
      "source": "src/packing-and-the-reverse-defect.md",
      "source_sha256": "62c22c6b1b66b169f4a4e063b5bb86d1189e75e562dded4bc2dda6bffb42fa0f",
      "status": "author_self_checked"
    },
    {
      "id": "polynomial-pairings-on-punctured-spaces",
      "title": "Polynomial pairings on punctured spaces",
      "reader": "polynomial-pairings-on-punctured-spaces.html",
      "source": "src/polynomial-pairings-on-punctured-spaces.md",
      "source_sha256": "ce87725e372d939dadd969a7dbac16c7fd24bd460e81e7c2a27e4d3dfb6da6f1",
      "status": "author_self_checked"
    },
    {
      "id": "direction-spaces-and-compatible-holes",
      "title": "Direction spaces and compatible holes",
      "reader": "direction-spaces-and-compatible-holes.html",
      "source": "src/direction-spaces-and-compatible-holes.md",
      "source_sha256": "e377c830e5e55ddc2929d7d72d0d434a83d468ba66aa2933a41c2b31fb9c3234",
      "status": "author_self_checked"
    },
    {
      "id": "random-slabs-and-the-local-lemma",
      "title": "Random slabs and the local lemma",
      "reader": "random-slabs-and-the-local-lemma.html",
      "source": "src/random-slabs-and-the-local-lemma.md",
      "source_sha256": "52d47d3ff24e29e05fa656fbb2206623290316a124922f9f443f22a2877f6483",
      "status": "author_self_checked"
    },
    {
      "id": "line-partitions-girth-and-the-counterexample",
      "title": "Line partitions, girth and the counterexample",
      "reader": "line-partitions-girth-and-the-counterexample.html",
      "source": "src/line-partitions-girth-and-the-counterexample.md",
      "source_sha256": "8f9f64b0dd4128cc929af59c246d9e219a2f025f35294e03d71cef9b1ff2eab1",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "programme lesson The regular trace and spectral phase (course the-kervaire-conjecture), Section 1 and Lemma 1.3: the trace on matrices over the group von Neumann algebra",
    "core course Complex Analysis (Lebl, Guide to Cultivating Complex Analysis): Theorem 3.3.11, the fundamental theorem of algebra",
    "core courses Abstract Algebra I and II (Judson, Abstract Algebra: Theory and Applications): Theorems 6.10, 6.19, 22.6, 22.7 and Corollaries 22.11, 22.12",
    "core course Probability (Grinstead-Snell, Introduction to Probability): Theorem 8.1, Chebyshev's inequality",
    "core course Discrete Mathematics (Levin, Discrete Mathematics: An Open Introduction): Section 4.3, Euler's formula"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}