# Open problems from the 1980s > 60 AI-made results on questions first raised the 1980s, each explained in plain English. Results on questions first raised the 1980s, going by the… Canonical: https://closedform.nishachay.tech/open-since/1980s/ Last updated: 2026-10-14 Results on questions first raised the 1980s, going by the earliest step each entry records. 60 AI-made results on questions first raised the 1980s, from 128 papers in the OpenAI math collection, each explained in plain English. 34 have a formal Lean proof listed; 26 are claimed in a paper and not yet machine-checked. They span 16 fields, led by differential geometry (10), algebraic and complex geometry (5) and partial differential equations (5). The oldest question here dates back to 1980: [The exact Sobolev endpoint for Schrödinger convergence](https://closedform.nishachay.tech/e/openai-math-2026-080/). Results rated Landmark include [Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity](https://closedform.nishachay.tech/e/openai-math-2026-034/) and [The Falconer distance conjecture](https://closedform.nishachay.tech/e/openai-math-2026-073/). ## Results (60) - [Milne’s rationality conjecture and algebraic specialization](https://closedform.nishachay.tech/e/openai-math-2026-001/) - Entry 001 · After reduction mod p, Hodge classes on abelian varieties are claimed to give one rational number in every cohomology. (Claimed, not checked) - [Torus-packet equidistribution in prime, quartic, and sextic degrees](https://closedform.nishachay.tech/e/openai-math-2026-015/) - Entry 015 · Number-field grids spread evenly over all grid shapes in prime degrees from five, and in degrees four and six. (Formal proof listed) - [The local p -adic section conjecture and global consequences](https://closedform.nishachay.tech/e/openai-math-2026-019/) - Entry 019 · Points on p-adic curves of genus two or more are claimed to match exactly the sections of their fundamental group. (Claimed, not checked) - [Uchida’s conjecture for open homomorphisms of Galois groups](https://closedform.nishachay.tech/e/openai-math-2026-031/) - Entry 031 · Every open map between these Galois groups is claimed to come from a unique map between the fields. (Claimed, not checked) - [Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity](https://closedform.nishachay.tech/e/openai-math-2026-034/) - Entry 034 · A family of 14 papers claims log abundance in characteristic zero and the effective Iitaka fibration conjecture. (Claimed, not checked) - [Log-canonical threefold abundance in numerical dimension one](https://closedform.nishachay.tech/e/openai-math-2026-035/) - Entry 035 · In characteristic above 3, a nef log canonical threefold divisor of numerical dimension one is claimed to give a map. (Claimed, not checked) - [Fujita’s freeness conjecture](https://closedform.nishachay.tech/e/openai-math-2026-038/) - Entry 038 · On every smooth shape, n + 1 copies of a positive bundle are always enough for Fujita's freeness. (Claimed, not checked) - [Termination of projective and Kähler fourfold minimal model programs](https://closedform.nishachay.tech/e/openai-math-2026-056/) - Entry 056 · Every permitted minimal model program on a projective log canonical fourfold claims to stop after finitely many steps. (Claimed, not checked) - [The generalized Mukai conjecture](https://closedform.nishachay.tech/e/openai-math-2026-063/) - Entry 063 · A claimed proof that rational curves on a Fano manifold cap its Picard number, with equality only for product spaces. (Claimed, not checked) - [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) - [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) - [Subpolynomial dimension reduction in L p](https://closedform.nishachay.tech/e/openai-math-2026-094/) - Entry 094 · Points in Lp space can be squeezed into far fewer coordinates if distances may stretch slightly. (Formal proof listed) - [The Euclidean Steinitz–Bergström bound](https://closedform.nishachay.tech/e/openai-math-2026-097/) - Entry 097 · One choice of signs keeps every running total of unit vectors within a constant times √d, at any length. (Formal proof listed) - [The sharp simplex conjecture for isotropic constants](https://closedform.nishachay.tech/e/openai-math-2026-101/) - Entry 101 · In every dimension, simplices alone have the largest isotropic constant among convex bodies, the paper claims. (Claimed, not checked) - [A cubic permanent–determinant lower bound](https://closedform.nishachay.tech/e/openai-math-2026-108/) - Entry 108 · Disguising the permanent as a determinant needs a matrix that grows at least like the cube of its size. (Formal proof listed) - [Beyond the square-root exponent for depth-three circuits](https://closedform.nishachay.tech/e/openai-math-2026-112/) - Entry 112 · One fast-to-decide problem needs three-layer circuits bigger than any 2^(A√n) for large inputs. (Formal proof listed) - [Approximate counting and entropy of perfect matchings](https://closedform.nishachay.tech/e/openai-math-2026-113/) - Entry 113 · A fast coin-flipping method estimates the number of perfect matchings in any network, to any chosen accuracy. (Formal proof listed) - [A factor-two approximation for shortest common superstring](https://closedform.nishachay.tech/e/openai-math-2026-128/) - Entry 128 · A fast method builds a string containing all given pieces, never more than twice the shortest possible. (Formal proof listed) - [Classwise permanence for weakly reversible mass-action systems](https://closedform.nishachay.tech/e/openai-math-2026-149/) - Entry 149 · With fixed rates, every weakly reversible chemical network herds each class of states into one common window. (Formal proof listed) - [Weak mixing of triangular billiards with an irrational angle](https://closedform.nishachay.tech/e/openai-math-2026-150/) - Entry 150 · A paper claims billiards in every triangle with an irrational angle forget their start: they are weakly mixing. (Formal proof listed) - [Superexponential van der Waerden numbers](https://closedform.nishachay.tech/e/openai-math-2026-160/) - Entry 160 · Proves van der Waerden numbers grow faster than any exponential, answering a question of Erdős for two colors. (Formal proof listed) - [Deterministic nonbipartite Ramanujan graphs in every fixed degree](https://closedform.nishachay.tech/e/openai-math-2026-178/) - Entry 178 · A fixed recipe builds networks with best-possible mixing, in every degree and every large even size. (Claimed, not checked) - [Correspondence coloring with a fixed forbidden subgraph](https://closedform.nishachay.tech/e/openai-math-2026-184/) - Entry 184 · Networks with no fixed-size clique have larger independent sets and need fewer colors, by a log factor. (Formal proof listed) - [The blockwise Alperin weight conjecture](https://closedform.nishachay.tech/e/openai-math-2026-202/) - Entry 202 · A paper claims to prove Alperin's weight conjecture, block by block, for every finite group and every prime. (Claimed, not checked) - [Donovan's conjecture over fields and complete mixed-characteristic DVRs](https://closedform.nishachay.tech/e/openai-math-2026-203/) - Entry 203 · Two papers claim Donovan's conjecture: blocks with bounded defect fall into finitely many types, for every prime. (Claimed, not checked) - [Finite lattice representation and undecidability](https://closedform.nishachay.tech/e/openai-math-2026-206/) - Entry 206 · Some finite lattices are not the congruence lattice of any finite algebra, and no program can tell which. (Formal proof listed) - [Directional zero–one laws beyond iid environments and iid ballisticity](https://closedform.nishachay.tech/e/openai-math-2026-220/) - Entry 220 · A walk in a random environment escapes in a given direction with probability 0 or 1, and escape forces positive speed. (Formal proof listed) - [Perceptron free energies and microscopic jamming exponents](https://closedform.nishachay.tech/e/openai-math-2026-222/) - Entry 222 · Exact free energies for Ising and spherical perceptrons at every density, and the gap and force exponents at jamming. (Formal proof listed) - [Gaussian free field limits throughout the balanced six-vertex regime](https://closedform.nishachay.tech/e/openai-math-2026-225/) - Entry 225 · The balanced six-vertex height surface tends to a Gaussian free field for all c up to 2, with variance 1/arcsin(c/2). (Claimed, not checked) - [The three-quarter exponent for honeycomb self-avoiding walk](https://closedform.nishachay.tech/e/openai-math-2026-237/) - Entry 237 · Claims random self-avoiding walks of length n on the honeycomb lattice span about n^(3/4), the long-predicted exponent. (Formal proof listed) - [A torsion-free hyperbolic group that is neither residually finite nor linear over any field](https://closedform.nishachay.tech/e/openai-math-2026-252/) - Entry 252 · A torsion-free hyperbolic group in which some element vanishes in every finite quotient. (Formal proof listed) - [Quasi-isometric recognition of virtually polycyclic groups](https://closedform.nishachay.tech/e/openai-math-2026-255/) - Entry 255 · Any finitely generated group that looks like a virtually polycyclic group from far away is virtually polycyclic. (Formal proof listed) - [A hyperbolic group without a geometric CAT(0) action](https://closedform.nishachay.tech/e/openai-math-2026-257/) - Entry 257 · A hyperbolic group that cannot act geometrically on any CAT(0) space, in any dimension. (Formal proof listed) - [The ionization and generalized ionization conjectures](https://closedform.nishachay.tech/e/openai-math-2026-263/) - Entry 263 · A nucleus can hold only a fixed number of extra electrons, however large its charge. (Formal proof listed) - [The spin-one Haldane gap](https://closedform.nishachay.tech/e/openai-math-2026-268/) - Entry 268 · Claims the spin-one Haldane gap: the Heisenberg chain on even rings has an energy gap that stays above log(20)/784. (Claimed, not checked) - [Uniform Laughlin gap and stability under bounded scalar disorder](https://closedform.nishachay.tech/e/openai-math-2026-269/) - Entry 269 · Proves a uniform energy gap of at least 1/25 above the Laughlin state on the sphere; claims it survives weak disorder. (Formal proof listed) - [From scale symmetry to local conformal symmetry in four-dimensional QFT](https://closedform.nishachay.tech/e/openai-math-2026-282/) - Entry 282 · Claims that scale symmetry forces local conformal symmetry for a broad class of four-dimensional quantum field theories. (Claimed, not checked) - [Counterexamples to Baum–Connes and Kadison–Kaplansky](https://closedform.nishachay.tech/e/openai-math-2026-285/) - Entry 285 · Claims torsion-free groups that break the Kadison–Kaplansky projection conjecture and reduced Baum–Connes. (Claimed, not checked) - [Rigidity and arithmetic of lattice von Neumann algebras](https://closedform.nishachay.tech/e/openai-math-2026-286/) - Entry 286 · Claims the von Neumann algebra of a property (T) lattice remembers its group, its twist and its scale. (Claimed, not checked) - [Relative bicentralizers and modular spectral recovery](https://closedform.nishachay.tech/e/openai-math-2026-290/) - Entry 290 · Claims every expected inclusion of von Neumann algebras holds an amenable piece with the same commutant in the core. (Formal proof listed) - [Four-dimensional disk embedding and Wall's conjecture](https://closedform.nishachay.tech/e/openai-math-2026-305/) - Entry 305 · Claims the two-generator free group is not good for 4D surgery, plus a counterexample to Wall's question. (Claimed, not checked) - [The Grothendieck homotopy hypothesis](https://closedform.nishachay.tech/e/openai-math-2026-312/) - Entry 312 · Grothendieck's algebraic ∞-groupoids capture all shapes of spaces, a paper claims, with its key step checked in Lean. (Formal proof listed) - [Thomason model structures in all strict higher dimensions](https://closedform.nishachay.tech/e/openai-math-2026-317/) - Entry 317 · Strict n-categories carry a Thomason model structure in every dimension, settling an Ara–Maltsiniotis conjecture. (Formal proof listed) - [Tingley’s sphere-isometry problem](https://closedform.nishachay.tech/e/openai-math-2026-322/) - Entry 322 · Every distance-keeping map between unit spheres of two real Banach spaces extends to a linear map of the spaces. (Formal proof listed) - [The cotype–cotype conjecture under the approximation property](https://closedform.nishachay.tech/e/openai-math-2026-326/) - Entry 326 · For Banach spaces with the approximation property, K-convexity equals finite cotype of the space and its dual. (Formal proof listed) - [Gromov’s integral scalar-curvature bound for simplicial volume](https://closedform.nishachay.tech/e/openai-math-2026-335/) - Entry 335 · Claims Gromov's integral scalar-curvature bound for simplicial volume and the Gromov–Lawson aspherical conjecture. (Claimed, not checked) - [Spectral scalar curvature, Urysohn width, and macroscopic dimension](https://closedform.nishachay.tech/e/openai-math-2026-336/) - Entry 336 · Claims Gromov's codimension-two width bound for positive scalar curvature in every dimension, and a spectral version. (Claimed, not checked) - [Yau's uniformization conjecture](https://closedform.nishachay.tech/e/openai-math-2026-338/) - Entry 338 · Claims to prove Yau's uniformization conjecture: positive bisectional curvature makes a complete Kähler manifold ℂⁿ. (Claimed, not checked) - [Katok's entropy rigidity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-339/) - Entry 339 · Claims to prove Katok's entropy rigidity conjecture for negatively curved manifolds of dimension three and higher. (Claimed, not checked) - [A counterexample to the nearby Lagrangian conjecture](https://closedform.nishachay.tech/e/openai-math-2026-340/) - Entry 340 · Claims a counterexample to Arnold's nearby Lagrangian conjecture, in the cotangent bundle of a product of two spheres. (Claimed, not checked) - [Symplectic ball packing in higher dimensions](https://closedform.nishachay.tech/e/openai-math-2026-343/) - Entry 343 · Proves Siegel and Yao's conjecture: from dimension six, ball packings are limited only by volume and the two-ball test. (Formal proof listed) - [Counterexamples to stable-Morse and strong Arnold fixed-point bounds](https://closedform.nishachay.tech/e/openai-math-2026-347/) - Entry 347 · Claims counterexamples to strong forms of Arnold's fixed-point conjecture; the three-point quadric example is in Lean. (Formal proof listed) - [Yau’s nodal bounds: surfaces and higher dimensions](https://closedform.nishachay.tech/e/openai-math-2026-350/) - Entry 350 · Proves Yau's sharp nodal-length bound on smooth surfaces, and shows it fails for smooth metrics in dimensions 3 to 5. (Formal proof listed) - [A finite-time singularity of Calabi flow](https://closedform.nishachay.tech/e/openai-math-2026-352/) - Entry 352 · Claims a smooth starting metric on complex projective 10-space whose Calabi flow breaks down in finite time. (Claimed, not checked) - [Failure of integer-degree harmonic dimension comparison](https://closedform.nishachay.tech/e/openai-math-2026-361/) - Entry 361 · Proves Yau's Euclidean bound on polynomial-growth harmonic functions fails under nonnegative Ricci curvature. (Formal proof listed) - [Global smoothness for relativistic Vlasov–Maxwell](https://closedform.nishachay.tech/e/openai-math-2026-362/) - Entry 362 · Proves the 3D relativistic Vlasov–Maxwell plasma equations have smooth solutions for all time, from any admissible data. (Formal proof listed) - [Nonuniqueness with local conservation for the hard-sphere Boltzmann equation](https://closedform.nishachay.tech/e/openai-math-2026-363/) - Entry 363 · Proves the periodic hard-sphere Boltzmann equation can have two different solutions from the same initial gas. (Formal proof listed) - [Joint metric and connection recovery from one boundary patch](https://closedform.nishachay.tech/e/openai-math-2026-365/) - Entry 365 · Claims one boundary patch determines a smooth metric; proves rough 3D conductivities can share all boundary data. (Formal proof listed) - [The planar Mumford–Shah regularity conjecture and local weak- L 4 gradient bounds](https://closedform.nishachay.tech/e/openai-math-2026-366/) - Entry 366 · Claims interior Mumford–Shah regularity in the plane: edges are smooth arcs, crack tips or 120-degree triple junctions. (Claimed, not checked) - [The critical dimension for the one-phase Bernoulli problem](https://closedform.nishachay.tech/e/openai-math-2026-367/) - Entry 367 · Claims seven is the critical dimension for one-phase Bernoulli minimizers, so free boundaries are smooth through six. 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