{
  "id": "strict-hot-spots-on-simply-connected-planar-domains",
  "title": "Strict hot spots on simply connected planar domains",
  "description": "On a bounded simply connected planar domain with smooth boundary, no eigenfunction of the first positive Neumann eigenvalue has an interior critical point, so its maximum and minimum are attained only on the boundary: Burdzy's simply connected hot-spots conjecture, in strict form and for every eigenfunction. The proof: Filonov's gap between the Neumann and Dirichlet eigenvalues; a conformal map smooth up to the boundary, boundary signs and the multiplicity bound; Rohleder's variational principle for tangent vector fields; boundary kernels of the disc with a subcritical potential and a multiplier identity; conditional negativity of a reciprocal kernel through matrices with one positive eigenvalue; a Hilbert-space embedding of the circle. Standard facts on Sobolev spaces and elliptic regularity are named hypotheses. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "neumann-eigenfunctions-and-the-hot-spots-problem",
      "title": "Neumann eigenfunctions and the hot-spots problem",
      "reader": "neumann-eigenfunctions-and-the-hot-spots-problem.html",
      "source": "src/neumann-eigenfunctions-and-the-hot-spots-problem.md",
      "source_sha256": "079f44eeac676afcfceaaa6c54466f791983ccd1401db80813ccc7a6c03b6ca2",
      "status": "author_self_checked"
    },
    {
      "id": "conformal-coordinates-hopfs-lemma-and-boundary-signs",
      "title": "Conformal coordinates, Hopf's lemma and boundary signs",
      "reader": "conformal-coordinates-hopfs-lemma-and-boundary-signs.html",
      "source": "src/conformal-coordinates-hopfs-lemma-and-boundary-signs.md",
      "source_sha256": "078fa00a20eb40df90c87d8c3acbf24fa444f390a38078be7c75ff4225bb3d5d",
      "status": "author_self_checked"
    },
    {
      "id": "a-variational-principle-for-tangent-vector-fields",
      "title": "A variational principle for tangent vector fields",
      "reader": "a-variational-principle-for-tangent-vector-fields.html",
      "source": "src/a-variational-principle-for-tangent-vector-fields.md",
      "source_sha256": "d5f216ef129f7de5f0e5ed961f55c4a7b4ef31a385a627a346d13df4cfb1fb43",
      "status": "author_self_checked"
    },
    {
      "id": "boundary-kernels-and-the-multiplier-identity",
      "title": "Boundary kernels and the multiplier identity",
      "reader": "boundary-kernels-and-the-multiplier-identity.html",
      "source": "src/boundary-kernels-and-the-multiplier-identity.md",
      "source_sha256": "cc6bfbc07a7fc565acb5d37321c1b567272bc917f3451ca333debe9edc5f38a0",
      "status": "author_self_checked"
    },
    {
      "id": "reciprocal-kernels-and-conditional-negativity",
      "title": "Reciprocal kernels and conditional negativity",
      "reader": "reciprocal-kernels-and-conditional-negativity.html",
      "source": "src/reciprocal-kernels-and-conditional-negativity.md",
      "source_sha256": "935ec17625619f322056943f5bbf255cc92eb433aef46a52d8adac3175a9e54b",
      "status": "author_self_checked"
    },
    {
      "id": "the-absence-of-interior-critical-points",
      "title": "The absence of interior critical points",
      "reader": "the-absence-of-interior-critical-points.html",
      "source": "src/the-absence-of-interior-critical-points.md",
      "source_sha256": "29ceb71c138f1046b820d946748461fb49a7fffb39b070329fa79bb6e7651da6",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course Elliptic operators and boundary problems, lesson Boundary energy, local inverses, and harmonic data: the energy solution of the Dirichlet problem (B10), the Poincare inequality (B11), the zero-trace space (B6), and H^2 and smooth regularity of the Dirichlet problem (B29)",
    "course Constant-coefficient equations and solvability, lesson Anisotropic derivative classes and analyticity: Theorem 3.1 (distributional solutions of elliptic equations with constant coefficients are real analytic)",
    "course The Riemann zeta function, lesson Dirichlet series and Euler products: Appendix A, Lemmas A.2 and A.3 (primitives on simply connected domains, isolated zeros, the harmonic maximum principle)",
    "course Foundations of von Neumann algebras, lesson Hilbert spaces and compact operators: the projection theorem and the Riesz-Frechet theorem (Theorems 2.2 and 2.3)",
    "the named hypotheses (SOB) and (ER) of the first lesson; the lessons prove every other fact they use"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}