Open problems from the 1990s
Results on questions first raised the 1990s, going by the earliest step each entry records.
55 AI-made results on questions first raised the 1990s, from 97 papers in the OpenAI math collection, each explained in plain English. 30 have a formal Lean proof listed; 25 are claimed in a paper and not yet machine-checked. They span 15 fields, led by theoretical computer science (7), combinatorics (6) and probability and statistical mechanics (6). The oldest question here dates back to 1990: The Deligne–Drinfeld conjecture. Results rated Landmark include Modularity of elliptic curves over imaginary quadratic fields and Local smoothing in three dimensions.
Number theory4
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The Deligne–Drinfeld conjecture
Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture.
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Function-field reconstruction from Milnor K-theory and Galois data
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from , , and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic.
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Unrestricted pro-modularity at the prime two
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.
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Modularity of elliptic curves over imaginary quadratic fields
Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place.
Claimed, not checked
Algebraic and complex geometry5
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Projective contact classification and the LeBrun–Salamon conjecture
Proves the LeBrun–Salamon conjecture: every closed connected positive quaternionic-Kähler manifold of real dimension at least eight is homothetic to a compact symmetric Wolf space. It also proves contact-Fano homogeneity and classifies smooth connected complex projective contact manifolds of complex dimension at least three: those with are adjoint varieties with their canonical contact structures, while those with have underlying manifold for a smooth projective variety Z.
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Virasoro constraints for complete intersections and projective-bundle towers
Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.
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Bounded klt complements for Fano contractions
Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational ϵ.
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The Campana–Peternell conjecture in dimension six
Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.
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Anticanonical nonvanishing in every dimension
If X is a smooth connected complex projective variety and admits a smooth Hermitian metric with nonnegative curvature, then for some m > 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.
Claimed, not checked
Real and complex analysis3
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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 , with Sobolev regularity above . In particular, the critical L3 estimate holds with every positive Sobolev loss.
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Riesz transforms and rectifiability in higher codimension
Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and , an n-Ahlfors–David regular Radon measure on ℝd 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.
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An L3 bound for the trilinear Hilbert transform
Proves that the principal-value trilinear Hilbert transform with shifts , , is bounded from to . This resolves the L3 exponent case of the standard conjecture for slopes 1, 2, 3.
Claimed, not checked
Theoretical computer science7
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Quasipolynomial algorithms for mean-payoff, stochastic and parity games
Gives deterministic algorithms using bit operations, for complete binary input length L, for ordinary mean-payoff games and two separate extensions. They compute exact values and optimal positional strategies in ordinary games, the nonnegative expectation-of-liminf value set in turn-based stochastic games, and the winning set for nonnegative liminf mean payoff conjoined with parity. Signed rewards, rational chance probabilities, and parity priorities are unrestricted and binary-encoded.
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Bin packing and unbounded configuration-LP gaps
Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items.
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Exponential semidefinite complexity of perfect matching
Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift , allowing arbitrary real positive semidefinite factors.
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Average sensitivity of polynomial threshold functions
Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most , uniformly for . Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with .
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Rapid mixing of graph switches for every degree sequence
Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time.
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A superquadratic separation of sensitivity and block sensitivity
Constructs total Boolean functions with block sensitivity for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together.
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Homogeneous depth-five lower bounds for iterated matrix multiplication
Over every characteristic-zero field, the entry of a product of n independent variable matrices requires gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.
Formal proof listed
Dynamical systems and ergodic theory1
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Pointwise multiple ergodic averages for mixing transformations
Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.
Claimed, not checked
Combinatorics6
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A counterexample to periodic tiling in dimension three
Constructs a finite translational tile in ℤ3 that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ3, even with arbitrary real translation vectors.
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Borsuk's conjecture fails in dimension nine
Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ4, with the Frobenius metric.
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Counterexamples to Sidorenko’s conjecture and the forcing conjecture
Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness.
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Combinatorial invariance of Kazhdan–Lusztig polynomials
Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems.
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Shareshian–Wachs elementary positivity
Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in . The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity.
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Seymour’s second-neighborhood conjecture
Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs.
Formal proof listed
Algebra1
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Eisenbud–Green–Harris and lex-plus-powers
Proves the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field. Any homogeneous ideal containing a regular sequence, of arbitrary length and degrees at least two, admits a lex-plus-powers ideal with the same Hilbert function and no smaller graded Betti numbers.
Claimed, not checked
Probability and statistical mechanics6
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Critical percolation on every quasi-transitive graph
Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with : at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ3.
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The Benjamini–Schramm nonuniqueness conjecture
Proves for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition.
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The low-temperature Sherrington–Kirkpatrick fluctuation law
For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to , with , confirming the predicted n1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.
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GOE bulk universality for regular graphs with weak Anderson disorder
For every fixed degree d ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random d-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throughout compact bands strictly inside the clean spectral edges.
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Critical and quenched near-critical universality for Poisson–Voronoi percolation
Proves Cardy's formula for annealed critical Poisson–Voronoi crossing probabilities in every bounded Jordan quadrilateral. With each model normalized by its own expected unit-square pivotal count, the conditional joint near-critical crossing-threshold laws for rational polygonal quads converge in environment probability to the triangular-lattice reference law. This establishes quenched near-critical universality for crossing thresholds.
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Limiting random SAT thresholds, sharp variance and computability
For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance for every fixed k ≥ 3, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent [ECCC TR26-229](https://eccc.weizmann.ac.il/report/2026/229/), made public October 5, 2026; this family supplies another proof and the sharper variance and computability results.
Formal proof listed
Mathematical logic2
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Separating choiceless counting from polynomial time and witnessed choice
Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank.
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Weak normalization implies strong normalization in pure type systems
Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates. This resolves the β-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts.
Formal proof listed
Group theory3
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Cannon's conjecture
Every word-hyperbolic group with boundary homeomorphic to S2 admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold.
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Gersten’s conjecture and virtual compact specialness of one-relator groups
Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup , with , is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.
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A group without fixed price
Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away from one, while a sequence of finite height extensions has costs tending to one.
Claimed, not checked
Mathematical physics5
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Exactly three mutually unbiased bases in dimension six
Proves , resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ6, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class.
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Threshold and positive-energy bound states of the BFSS matrix model
Proves that the undeformed relative BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For , a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper's exclusion of additional bound states at N = 2.
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Parity is not in QAC0
Resolves Moore's parity conjecture in the measured-output model: constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates and polynomially many total qubits cannot compute parity with any fixed positive worst-case advantage. Ancillas start in zero, one output qubit is measured, and all other registers may be discarded. Xu–Li's reductions give the same bounded-error obstruction for strict majority.
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Threshold repetition for entangled games
Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction of k independent repetitions decays exponentially in k, for . Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.
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Exact quantum factoring over a fixed finite gate set
Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices.
Formal proof listed
Operator algebras2
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Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability
Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.
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Two notions of free entropy differ even when both are finite
Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy . This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.
Formal proof listed
Topology5
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The purely cosmetic surgery conjecture
Distinct Dehn surgery slopes on a nontrivial smooth knot in S3 never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.
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Failure of rational injectivity for maximal coarse assembly
Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture.
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The Hovey–Strickland and Chai conjectures
Proves Chai's invariant-ideal conjecture for Lubin–Tate deformation rings over finite residue fields, at every prime and positive height n. Through the implication of Barthel–Heard–Naumann, this proves the Hovey–Strickland conjecture: dualizable -local spectra have exactly thick tensor ideals, and their Balmer spectrum is a chain of points.
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Finite generation for the -local sphere
Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the -local sphere is a finitely generated ℤp-module, for every prime, positive height and integer degree. The same conclusion holds after -localizing any finite p-local spectrum.
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Chromatic splitting: filtrations and counterexamples
Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and (p ≥ 7). Nevertheless, for n ≥ 1 and , the overlap admits a -stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of -local modules over the derived completed sphere.
Claimed, not checked
Functional analysis1
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Metric Markov cotype of ℓ1 and Hilbert-space Lipschitz extension
Proves that real ℓ1 has metric Markov cotype two, answering Mendel and Naor's question. Consequently, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ1 extends to the whole space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.
Claimed, not checked
Differential geometry4
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The Solomon–Yau least-volume conjecture
Proves the Solomon–Yau least-volume conjecture for minimal hypersurfaces of round spheres. For every m ≥ 2, a closed connected minimal immersion into the unit sphere with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.
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Scalar curvature and finite-time Ricci-flow singularities
Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.
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Unique tangent flows at the first surface singularity
Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ3. At every singular point, all fixed-center backward tangent flows agree as area measures at every negative time in the original ambient coordinates, without mean-convexity or a prescribed tangent model.
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Bi-Lipschitz coordinates at every regular RCD point
Proves that every regular point of a noncollapsed space, for K ∈ ℝ and integer n ≥ 2, has an open neighborhood bi-Lipschitz to an open subset of ℝn. Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly , and the chart compares ambient distances with a point-dependent finite constant.
Claimed, not checked