# Open problems from the 2000s

> 48 AI-made results on questions first raised the 2000s, each explained in plain English. Results on questions first raised the 2000s, going by the…

Canonical: https://closedform.nishachay.tech/open-since/2000s/
Last updated: 2026-10-14

Results on questions first raised the 2000s, going by the earliest step each entry records.

48 AI-made results on questions first raised the 2000s, from 98 papers in the OpenAI math collection, each explained in plain English. 34 have a formal Lean proof listed; 14 are claimed in a paper and not yet machine-checked. They span 14 fields, led by theoretical computer science (8), algebraic and complex geometry (6) and differential geometry (6). The oldest question here dates back to 2000: [Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII](https://closedform.nishachay.tech/e/openai-math-2026-042/). Results rated Landmark include [The Unique Games Conjecture and optimal approximation thresholds](https://closedform.nishachay.tech/e/openai-math-2026-102/) and [Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling](https://closedform.nishachay.tech/e/openai-math-2026-223/).

## Results (48)

- [Irrationality of Catalan’s constant](https://closedform.nishachay.tech/e/openai-math-2026-005/) - Entry 005 · Catalan's constant, 1 − 1/9 + 1/25 − 1/49 + …, is not a fraction. (Formal proof listed)
- [Zilber–Pink in abelian varieties and the Siegel threefold](https://closedform.nishachay.tech/e/openai-math-2026-016/) - Entry 016 · Unlikely intersections in abelian varieties are claimed to be finite in number, in every dimension. (Claimed, not checked)
- [Hyperkähler SYZ and projective-space bases](https://closedform.nishachay.tech/e/openai-math-2026-041/) - Entry 041 · Every nonzero nef isotropic line bundle on a compact hyperkähler manifold is claimed to give a torus fibration. (Claimed, not checked)
- [Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII](https://closedform.nishachay.tech/e/openai-math-2026-042/) - Entry 042 · A claimed proof that every complex K3 surface is Oka: local holomorphic maps into it can be copied by entire maps. (Claimed, not checked)
- [A stable-coordinate counterexample in four variables](https://closedform.nishachay.tech/e/openai-math-2026-049/) - Entry 049 · A polynomial that becomes a coordinate with one extra variable, but not without it, disproves two conjectures. (Formal proof listed)
- [Tangent splittings and product decompositions](https://closedform.nishachay.tech/e/openai-math-2026-052/) - Entry 052 · Two integrable families of directions on a compact Kähler manifold make its universal cover a product. (Formal proof listed)
- [Fundamental groups of special complex varieties and root orbifolds](https://closedform.nishachay.tech/e/openai-math-2026-057/) - Entry 057 · Special compact Kähler manifolds claim virtually abelian fundamental groups in every dimension, as Campana predicted. (Claimed, not checked)
- [Global quantum geometric Langlands at irrational level](https://closedform.nishachay.tech/e/openai-math-2026-069/) - Entry 069 · A claimed proof of quantum geometric Langlands duality on any smooth complex curve at every irrational level. (Claimed, not checked)
- [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)
- [Bounded-distortion L 1 embeddings of planar and bounded-treewidth graphs](https://closedform.nishachay.tech/e/openai-math-2026-089/) - Entry 089 · Every planar road network can be redrawn in taxicab distance with stretch below one fixed number. (Formal proof listed)
- [The Gaussian propeller conjecture in every dimension](https://closedform.nishachay.tech/e/openai-math-2026-096/) - Entry 096 · Three 120-degree wedges are the best way to cut a bell-curve cloud, in every dimension. (Formal proof listed)
- [Compact counterexamples to bi-Lipschitz dimension reduction](https://closedform.nishachay.tech/e/openai-math-2026-098/) - Entry 098 · Some uncrowded point sets in infinite dimensions cannot be copied faithfully into any finite dimension. (Formal proof listed)
- [The sharp exponential scale of edit-distance distortion](https://closedform.nishachay.tech/e/openai-math-2026-099/) - Entry 099 · Edit distance between words can be drawn as ℓ1 distance only with distortion of size exp(√(log d·log log d)). (Formal proof listed)
- [The Unique Games Conjecture and optimal approximation thresholds](https://closedform.nishachay.tech/e/openai-math-2026-102/) - Entry 102 · Telling almost-solvable puzzles from hopeless ones can be impossibly hard. (Formal proof listed)
- [Perfect completeness for 2-to-1 games](https://closedform.nishachay.tech/e/openai-math-2026-105/) - Entry 105 · Telling fully solvable 2-to-1 games from nearly worthless ones is NP-hard. (Formal proof listed)
- [Hardness of coloring three-colorable graphs](https://closedform.nishachay.tech/e/openai-math-2026-106/) - Entry 106 · Even a three-colorable graph can be NP-hard to tell from one with only tiny independent sets. (Formal proof listed)
- [Approximate counting of common integer polymatroid bases](https://closedform.nishachay.tech/e/openai-math-2026-114/) - Entry 114 · Shared solutions of two capacity rules can be counted approximately, at a cost set by digits, not size. (Formal proof listed)
- [Sampling and counting contingency tables with arbitrary margins](https://closedform.nishachay.tech/e/openai-math-2026-115/) - Entry 115 · Number grids with fixed row and column sums can be sampled fairly and counted fast, whatever their size. (Formal proof listed)
- [Uniform black-box noncommutative identity testing across characteristics](https://closedform.nishachay.tech/e/openai-math-2026-116/) - Entry 116 · One fixed set of matrices exposes every nonzero small formula whose variables do not commute. (Formal proof listed)
- [Quantitative trace-reconstruction bounds with a uniform decoder](https://closedform.nishachay.tech/e/openai-math-2026-122/) - Entry 122 · Rebuilding a string from randomly thinned copies needs more than polynomially many copies. (Formal proof listed)
- [The metric k -median approximation threshold and recovery](https://closedform.nishachay.tech/e/openai-math-2026-125/) - Entry 125 · The best possible guarantee for placing k facilities is exactly 1 + 2/e, assuming P ≠ NP. (Formal proof listed)
- [Deterministic construction of strong thin spanning trees](https://closedform.nishachay.tech/e/openai-math-2026-174/) - Entry 174 · Every k-connected network has a spanning tree that uses at most a fixed multiple of 1/k of every cut. (Formal proof listed)
- [The second Kahn–Kalai conjecture with an edge-count bound](https://closedform.nishachay.tech/e/openai-math-2026-176/) - Entry 176 · A random graph contains a target graph soon after simple expected counts say it could, up to a log factor. (Formal proof listed)
- [Tensor saturation for even spin groups](https://closedform.nishachay.tech/e/openai-math-2026-204/) - Entry 204 · For even spin groups, an invariant after stretching the weights implies one before, if the weights pass a parity test. (Claimed, not checked)
- [The Mézard–Parisi formula for diluted spin glasses](https://closedform.nishachay.tech/e/openai-math-2026-221/) - Entry 221 · The Mézard–Parisi cavity formula gives the exact free energy of sparse spin glasses in the Panchenko–Talagrand class. (Formal proof listed)
- [Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling](https://closedform.nishachay.tech/e/openai-math-2026-223/) - Entry 223 · Critical random-cluster interfaces on the square grid converge to SLE curves for every cluster weight up to 4. (Claimed, not checked)
- [Exact Hausdorff gauges for SLE](https://closedform.nishachay.tech/e/openai-math-2026-230/) - Entry 230 · An explicit size rule gives every piece of an SLE random curve a positive, finite Hausdorff measure. (Formal proof listed)
- [Gaussian fields and interfaces for triangular-lattice Lipschitz heights](https://closedform.nishachay.tech/e/openai-math-2026-232/) - Entry 232 · Random Lipschitz height surfaces on the triangular lattice converge to the Gaussian free field. (Claimed, not checked)
- [Optimal logarithmic mixing of the Thorp shuffle](https://closedform.nishachay.tech/e/openai-math-2026-238/) - Entry 238 · Claims the Thorp shuffle on 2^d cards randomizes the whole deck in order d shuffles, the best possible order. (Formal proof listed)
- [Sharp singularity rates for symmetric random sign matrices](https://closedform.nishachay.tech/e/openai-math-2026-239/) - Entry 239 · Claims a random symmetric ±1 matrix is singular with probability (1/2 + o(1))ⁿ, with a matching biased-sign rate. (Claimed, not checked)
- [Rigidity of the Turing degrees](https://closedform.nishachay.tech/e/openai-math-2026-241/) - Entry 241 · Claims every symmetry of the Turing degrees is trivial, settling the rigidity question. (Formal proof listed)
- [Area laws and tensor networks for two-dimensional gapped systems](https://closedform.nishachay.tech/e/openai-math-2026-265/) - Entry 265 · Claims the 2D area law: gapped ground states on a square grid hold entanglement proportional to boundary, not volume. (Claimed, not checked)
- [Entanglement without distillable secret key](https://closedform.nishachay.tech/e/openai-math-2026-272/) - Entry 272 · Claims an entangled state in local dimension ten that yields no secret key, answering a 2005 open problem. (Formal proof listed)
- [The entropy photon-number inequality](https://closedform.nishachay.tech/e/openai-math-2026-273/) - Entry 273 · Proves the entropy photon-number inequality for two independent finite-energy light inputs mixed on a beam splitter. (Formal proof listed)
- [Polynomial-time unitary synthesis from a Boolean oracle](https://closedform.nishachay.tech/e/openai-math-2026-283/) - Entry 283 · Claims a polynomial-size quantum circuit can carry out any n-qubit operation, given a suitable Boolean oracle. (Claimed, not checked)
- [Kirchberg's O₂ norm-ultrapower embedding problem](https://closedform.nishachay.tech/e/openai-math-2026-292/) - Entry 292 · Proves, with a Lean check, an explicit C*-algebra that fits in no nuclear norm ultrapower, answering Kirchberg. (Formal proof listed)
- [A ZFC counterexample to Naimark's problem](https://closedform.nishachay.tech/e/openai-math-2026-297/) - Entry 297 · An algebra with one irreducible representation that is not the compact operators, built without extra axioms. (Formal proof listed)
- [Radius of comparison equals half the mean dimension](https://closedform.nishachay.tech/e/openai-math-2026-302/) - Entry 302 · Claims the radius of comparison of a minimal crossed product is half the mean dimension, the Phillips–Toms conjecture. (Claimed, not checked)
- [Weak pure infiniteness and Cuntz-algebra absorption](https://closedform.nishachay.tech/e/openai-math-2026-303/) - Entry 303 · Proves pure infiniteness implies strong pure infiniteness in all C*-algebras, part of a Kirchberg–Rørdam question. (Formal proof listed)
- [The complete Crouzeix conjecture](https://closedform.nishachay.tech/e/openai-math-2026-325/) - Entry 325 · The complete Crouzeix conjecture holds: matrix-coefficient polynomials of an operator obey the sharp factor 2. (Formal proof listed)
- [A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces](https://closedform.nishachay.tech/e/openai-math-2026-330/) - Entry 330 · Proves a uniformly discrete metric space can have a Lipschitz-free space lacking the bounded approximation property. (Formal proof listed)
- [Smooth isometric immersions of surfaces into ℝ 4](https://closedform.nishachay.tech/e/openai-math-2026-333/) - Entry 333 · Proves that every closed smooth curved surface can be placed in four-dimensional space with all its distances kept. (Formal proof listed)
- [Donaldson's hypersymplectic deformation conjecture](https://closedform.nishachay.tech/e/openai-math-2026-341/) - Entry 341 · Claims every hypersymplectic four-manifold deforms, with its cohomology classes fixed, to a hyperkähler triple. (Claimed, not checked)
- [Donaldson's tamed-to-compatible conjecture](https://closedform.nishachay.tech/e/openai-math-2026-342/) - Entry 342 · Proves Donaldson's tamed-to-compatible conjecture for tamed almost-complex structures on closed four-manifolds. (Formal proof listed)
- [Nonnegative-curvature Einstein classification and an L 2 topological gap](https://closedform.nishachay.tech/e/openai-math-2026-348/) - Entry 348 · Classifies positive Einstein four-manifolds with nonnegative curvature, and claims a small-energy topological gap. (Formal proof listed)
- [The isoperimetric profile of the cubic three-torus](https://closedform.nishachay.tech/e/openai-math-2026-354/) - Entry 354 · Proves the isoperimetric conjecture for the cubic flat three-torus: balls, tubes, slabs and their complements. (Formal proof listed)
- [Weak MTW curvature gives convexity and regular optimal transport](https://closedform.nishachay.tech/e/openai-math-2026-360/) - Entry 360 · Proves weak MTW curvature makes every injectivity domain convex, as Villani conjectured, and transport maps Hölder. (Formal proof listed)
- [The three-dimensional Ball–Evans approximation problem](https://closedform.nishachay.tech/e/openai-math-2026-368/) - Entry 368 · Claims to solve the Ball–Evans problem in 3D: every Sobolev homeomorphism is a strong limit of smooth diffeomorphisms. (Claimed, not checked)

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