Subjects

Real and complex analysis

16 results from OpenAI’s math release. 16 explained so far. OpenAI has published Lean proofs for 10: 7 of the main theorem, 3 of part of the claim.

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• 071 Koebe’s circle-domain conjecture

  Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture.

  2 papers ▶Episode, 6:22 OpenAI published a Lean proof of the main theorem.

• 072 Brennan's conjecture and the integral-means spectrum

  Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, $|\phi'|^s$ is area-integrable for $4/3\lt s\lt 4$. The sharp universal integral-means identity is $B_{\mathcal S}(t)=|t|-1$ for t ≤ −2. A strict bound $B_b(-1)\lt 1/4$ for bounded univalent functions disproves Kraetzer's prediction at that parameter.

  2 papers ▶Episode, 5:31 OpenAI published a Lean proof of the main theorem.

• 073 The Falconer distance conjecture

  Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set $E\subset\mathbb R^d$ with Hausdorff dimension greater than $d/2$ determines a set of Euclidean distances of positive Lebesgue measure.

  ▶Episode, 5:43 OpenAI published a Lean proof of the main theorem.

• 074 Kakeya in three and four dimensions

  Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps $L^3(\mathbb R^3)$ to $L^3(S^2)$ with norm $O_\varepsilon(\delta^{-\varepsilon})$ for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.

  2 papers ▶Episode, 7:34

• 075 The $L\log L$ Fourier-convergence conjecture

  Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in $L\log L(\mathbb T)$ converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the $L\log L$ scale.

  ▶Episode, 6:00 OpenAI published a Lean proof of the main theorem.

• 076 Real ultraflat Littlewood polynomials and unbounded binary merit factors

  Constructs polynomials with N consecutive coefficients in $\{-1,1\}$ whose modulus is $(1+o(1))\sqrt N$ uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.

  3 papers ▶Episode, 6:11 OpenAI published a Lean proof of part of the claim.

• 077 Fourier restriction for positively curved surfaces

  Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface $\Sigma\subset\mathbb R^3$, including surfaces with boundary, the extension operator is bounded from $L^p(\Sigma)$ to $L^p(\mathbb R^3)$ for every p > 3.

  2 papers ▶Episode, 5:49

• 078 The three-dimensional Bochner–Riesz conjecture

  Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on $L^p(\mathbb R^3)$ for every $1\le p\le\infty$ whenever $\delta\gt \max\{3|1/p-1/2|-1/2,0\}$.

  ▶Episode, 4:58

• 079 Local smoothing in three dimensions

  Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range $2\lt p\lt \infty$, with Sobolev regularity above $\max\{0,1-3/p\}$. In particular, the critical L³ estimate holds with every positive Sobolev loss.

  ▶Episode, 6:32

• 080 The exact Sobolev endpoint for Schrödinger convergence

  Proves almost-everywhere convergence $e^{it\Delta}f\to f$ as $t\downarrow0$ for every $f\in H^{n/(2(n+1))}(\mathbb R^n)$ and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H^1/3.

  2 papers ▶Episode, 6:02

• 081 Riesz transforms and rectifiability in higher codimension

  Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and $2\le n\le d-2$, an n-Ahlfors–David regular Radon measure on ℝ^d is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L²-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds.

  ▶Episode, 5:30 OpenAI published a Lean proof of the main theorem.

• 082 Annular variation and dyadic absolute bounds for the triangular Hilbert transform

  Proves maximal and annular r-variation bounds, for every r > 2, from complex $L^3(\mathbb R^2)\times L^3(\mathbb R^2)$ to $L^{3/2}(\mathbb R^2)$. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $L^3\times L^3\times L^3$ point.

  3 papers ▶Episode, 5:35 OpenAI published a Lean proof of part of the claim.

• 083 Hilbert transforms along Lipschitz directions

  Proves a uniform strong L² bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an L²-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale.

  ▶Episode, 5:48 OpenAI published a Lean proof of the main theorem.

• 084 The geometric case of the Erdős similarity conjecture

  For every fixed $q\in(0,1)$, constructs compact subsets of $[0,1]$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of $\{q^n:n\ge1\}$, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.

  2 papers ▶Episode, 5:27 OpenAI published a Lean proof of part of the claim.

• 085 Endpoint Sobolev regularity of centered disk averages

  Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real $f\in W^{1,1}(\mathbb R^2)$, the centered disk maximal function satisfies $\|\nabla Mf\|_1\le C\|\nabla f\|_1$ with an absolute constant. It belongs locally to $W^{1,1}$ and has a globally integrable weak gradient.

  ▶Episode, 6:40 OpenAI published a Lean proof of the main theorem.

• 086 An L³ bound for the trilinear Hilbert transform

  Proves that the principal-value trilinear Hilbert transform with shifts $x-t$, $x-2t$, $x-3t$ is bounded from $L^3(\mathbb R)^3$ to $L^1(\mathbb R)$. This resolves the L³ exponent case of the standard conjecture for slopes 1, 2, 3.

  ▶Episode, 6:05

Summary and paper titles from OpenAI’s catalog, used under the Apache License 2.0. Last checked October 19, 2026.