{
  "id": "coboundary-expanders-of-bounded-degree",
  "title": "Coboundary expanders of bounded degree",
  "description": "Coboundary expansion of simplicial complexes over the field with two elements, and the complexes of OpenAI's theorem (2026) that bounded-degree coboundary expanders exist in every dimension. Definitions, weights, the relation to cohomology, examples, the passage to skeletons; coset complexes of groups; the groups of polynomial matrices and their coset complexes, with bounded degree and growth. The theorem is derived from two named statements, an expansion inequality relative to the cocycles and the vanishing of cohomology, which this course does not prove.",
  "units": [
    {
      "id": "coboundary-expansion",
      "title": "Coboundary expansion",
      "reader": "coboundary-expansion.html",
      "source": "src/coboundary-expansion.md",
      "source_sha256": "6732abea7f9d8a4e7d93fb0df1cf47b3dc51739c55a03c2e38099198ae5144b2",
      "status": "author_self_checked"
    },
    {
      "id": "coset-complexes",
      "title": "Coset complexes",
      "reader": "coset-complexes.html",
      "source": "src/coset-complexes.md",
      "source_sha256": "5f24d82d2eb6a9b8d8cb453fee70987ab4952e33cff7599114e42e07de38225e",
      "status": "author_self_checked"
    },
    {
      "id": "the-groups-and-the-complexes",
      "title": "The groups and the complexes",
      "reader": "the-groups-and-the-complexes.html",
      "source": "src/the-groups-and-the-complexes.md",
      "source_sha256": "215fc990784729af0e84b99a786f590ae8a8f319f4ab88d2f4031092376bb4ad",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Linear Algebra (B40): vector spaces over a field, here the field with two elements",
    "core course Abstract Algebra I (C30): groups, subgroups, cosets, normal subgroups, quotient groups",
    "core course Linear Algebra (B40): matrices over a commutative ring, determinants; division with remainder for polynomials over a field",
    "the lessons prove every other fact they use"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}