# Real and complex analysis > 16 AI-made results in real and complex analysis, each explained in plain English with its trust label, Lean status and sources. Canonical: https://closedform.nishachay.tech/science/mathematics/real-and-complex-analysis/ Last updated: 2026-10-14 16 results in OpenAI Math 2026. 16 AI-made results in real and complex analysis, from 26 papers in the OpenAI math collection, each explained in plain English. 10 have a formal Lean proof listed; 6 are claimed in a paper and not yet machine-checked. The oldest question here dates back to 1908: [Koebe’s circle-domain conjecture](https://closedform.nishachay.tech/e/openai-math-2026-071/). Results rated Landmark include [The Falconer distance conjecture](https://closedform.nishachay.tech/e/openai-math-2026-073/) and [Kakeya in three and four dimensions](https://closedform.nishachay.tech/e/openai-math-2026-074/). ## Results (16) - [Koebe’s circle-domain conjecture](https://closedform.nishachay.tech/e/openai-math-2026-071/) - Entry 071 · Every region of the sphere, however many holes it has, can be reshaped so that each hole is round. (Formal proof listed) - [Brennan's conjecture and the integral-means spectrum](https://closedform.nishachay.tech/e/openai-math-2026-072/) - Entry 072 · Reshaping a region into a disk can never squeeze too hard: Brennan's 1978 conjecture is proved. (Formal proof listed) - [The Falconer distance conjecture](https://closedform.nishachay.tech/e/openai-math-2026-073/) - Entry 073 · Any closed, bounded set whose dimension is more than half that of its space has distances with positive total length. (Formal proof listed) - [Kakeya in three and four dimensions](https://closedform.nishachay.tech/e/openai-math-2026-074/) - Entry 074 · Claimed proofs: the Kakeya maximal conjecture in 3D, and full dimension for every Kakeya set in 4D. (Claimed, not checked) - [The L log L Fourier-convergence conjecture](https://closedform.nishachay.tech/e/openai-math-2026-075/) - Entry 075 · Fourier sums of every L log L function converge to it at almost every point of the circle. (Formal proof listed) - [Real ultraflat Littlewood polynomials and unbounded binary merit factors](https://closedform.nishachay.tech/e/openai-math-2026-076/) - Entry 076 · Polynomials with only +1 and −1 coefficients can be almost perfectly flat on the unit circle. (Formal proof listed) - [Fourier restriction for positively curved surfaces](https://closedform.nishachay.tech/e/openai-math-2026-077/) - Entry 077 · Claimed proofs of the full Fourier restriction range p > 3 for the sphere and for positively curved surfaces. (Claimed, not checked) - [The three-dimensional Bochner–Riesz conjecture](https://closedform.nishachay.tech/e/openai-math-2026-078/) - Entry 078 · A claimed proof of the Bochner–Riesz conjecture in three dimensions, in its strict-order form. (Claimed, not checked) - [Local smoothing in three dimensions](https://closedform.nishachay.tech/e/openai-math-2026-079/) - Entry 079 · A claimed proof of Sogge's local smoothing conjecture for waves in 3D space, including the critical case p = 3. (Claimed, not checked) - [The exact Sobolev endpoint for Schrödinger convergence](https://closedform.nishachay.tech/e/openai-math-2026-080/) - Entry 080 · Claimed proofs that Schrödinger waves return to rough data at the exact smoothness threshold, in dimensions n ≥ 2. (Claimed, not checked) - [Riesz transforms and rectifiability in higher codimension](https://closedform.nishachay.tech/e/openai-math-2026-081/) - Entry 081 · A bounded Riesz transform forces surfaces in higher codimension to contain big flat-like pieces. (Formal proof listed) - [Annular variation and dyadic absolute bounds for the triangular Hilbert transform](https://closedform.nishachay.tech/e/openai-math-2026-082/) - Entry 082 · A triangle-shaped singular integral stays bounded at the symmetric L³ point, whatever cutoffs each point picks. (Formal proof listed) - [Hilbert transforms along Lipschitz directions](https://closedform.nishachay.tech/e/openai-math-2026-083/) - Entry 083 · Hilbert transforms along any Lipschitz direction field in the plane obey one energy bound at short lengths. (Formal proof listed) - [The geometric case of the Erdős similarity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-084/) - Entry 084 · A set covering almost all of [0, 1] can dodge every copy of a geometric list like 1/2, 1/4, 1/8, … (Formal proof listed) - [Endpoint Sobolev regularity of centered disk averages](https://closedform.nishachay.tech/e/openai-math-2026-085/) - Entry 085 · Taking the best disk average at each point never multiplies a surface's total slope by more than a fixed factor. (Formal proof listed) - [An L 3 bound for the trilinear Hilbert transform](https://closedform.nishachay.tech/e/openai-math-2026-086/) - Entry 086 · A singular average of three shifted functions obeys an L³ × L³ × L³ to L¹ size bound, the paper claims. (Claimed, not checked) ## Related - [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 10 here - [Landmark results](https://closedform.nishachay.tech/significance/landmark/) - Significance · 7 here - [Named conjectures](https://closedform.nishachay.tech/topic/named-conjectures/) - Topic · 6 here - [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 6 here - [Open since the 1970s](https://closedform.nishachay.tech/open-since/1970s/) - Era · 4 here - [Open since the 1990s](https://closedform.nishachay.tech/open-since/1990s/) - Era · 3 here - [Theoretical computer science](https://closedform.nishachay.tech/science/mathematics/theoretical-computer-science/) - Field · 40 results - [Combinatorics](https://closedform.nishachay.tech/science/mathematics/combinatorics/) - Field · 37 results - [Algebraic and complex geometry](https://closedform.nishachay.tech/science/mathematics/algebraic-and-complex-geometry/) - Field · 36 results - [Mathematics](https://closedform.nishachay.tech/science/mathematics/) - [All topics](https://closedform.nishachay.tech/topics/) - Every way to browse