Landmark results
Results rated Landmark: answers to well-known, long-standing questions in their field.
49 AI-made results rated Landmark, from 109 papers in the OpenAI math collection, each explained in plain English. 27 have a formal Lean proof listed; 22 are claimed in a paper and not yet machine-checked. They span 14 fields, led by operator algebras (7), real and complex analysis (7) and algebra (5). The oldest question here dates back to 1896: The quasi-Riemann hypothesis. Results rated Landmark include The full BSD formula from low Selmer corank and Hilbert’s tenth problem over ℚ.
Number theory5
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The full BSD formula from low Selmer corank
Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.
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The quasi-Riemann hypothesis
Proves that every Dirichlet L-function, including , is zero-free in , resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over . A companion gives a different proof of the zero-free half-plane .
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Hilbert’s tenth problem over ℚ
Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.
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Primitive roots for every admissible integer base
Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least primes in every sufficiently large interval have primitive root a, with .
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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.
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Algebraic and complex geometry3
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The rational Hodge conjecture for CM abelian varieties
Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic.
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Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity
Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef ℚ-Cartier adjoint is semiample. It also proves projective log abundance over every algebraically closed field of characteristic zero and the effective Iitaka fibration conjecture for smooth projective varieties of nonnegative Kodaira dimension in that setting. A further result resolves the finite-rational-coefficient index conjecture for connected projective semi-log-canonical log Calabi–Yau pairs over such fields in each fixed dimension and for each fixed finite set of rational boundary coefficients, with a uniform index independent of the number of components.
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Nagata’s conjecture and maximal Seshadri constants
Proves Nagata's strict inequality for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal multipoint Seshadri constants for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic.
Formal proof listed
Real and complex analysis7
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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.
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The Falconer distance conjecture
Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set with Hausdorff dimension greater than determines a set of Euclidean distances of positive Lebesgue measure.
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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 to with norm for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.
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The Fourier-convergence conjecture
Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the scale.
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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 , including surfaces with boundary, the extension operator is bounded from to for every p > 3.
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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 for every whenever .
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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.
Claimed, not checked
Convex and metric geometry1
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The Mahler conjectures, functional inequalities and polar-product symplectic width
Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product in dimension has Gromov width 4.
Formal proof listed
Theoretical computer science3
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The Unique Games Conjecture and optimal approximation thresholds
Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results.
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Exact derandomization of logarithmic space:
Proves , resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds.
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Matrix multiplication with exponent at most 9/4
Proves over ℂ, giving arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.
Formal proof listed
Dynamical systems and ergodic theory2
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Hilbert's sixteenth problem: uniform bounds for limit cycles
Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles.
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Rokhlin’s multiple-mixing problem
Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge.
Formal proof listed
Combinatorics4
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Graph coloring, clique minors, and Colin de Verdière invariants
Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy , where is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound . In the positive direction, every finite nonempty graph satisfies for a universal constant C.
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Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds
Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of with no nonconstant k-term progression has size at most , with positive constants depending only on k.
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Planar distinct distances and unit-distance bounds
Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but points of any n-point planar set determine at least distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by for an absolute δ > 0.
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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.
Formal proof listed
Algebra5
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A counterexample to Kaplansky’s zero-divisor conjecture
Constructs a finitely presented torsion-free group G whose group algebra has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.
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A torsion-free group algebra that is not directly finite
Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.
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The blockwise Alperin weight conjecture
Proves the numerical blockwise Alperin weight conjecture for every prime and every finite group: the number of irreducible Brauer characters in a block equals the number of conjugacy classes of its weights.
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Donovan's conjecture over fields and complete mixed-characteristic DVRs
Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p. Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.
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The ℓ¹-Bass conjecture for all discrete groups
Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.
Formal proof listed
Probability and statistical mechanics3
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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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Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling
Proves chordal SLEκ limits for critical square-lattice random-cluster interfaces for , with : bounded Jordan domains are allowed for q ≥ 1, and smooth Jordan domains for q < 1, under the stated marked-boundary approximations. For , complete nested plane loops converge to CLEκ.
Claimed, not checked
Operator algebras7
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Counterexamples to Baum–Connes and Kadison–Kaplansky
Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C∗-algebra, disproving the Kadison–Kaplansky conjecture.
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Isomorphism of the free group factors
Resolves the free group factor isomorphism problem: , and hence all interpolated free group factors, including , are isomorphic. Their common factor has fundamental group .
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Kadison's similarity conjecture
Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a -homomorphism after conjugation by a bounded invertible operator.
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Strong Kadison–Kastler stability and its spatial boundaries
Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C∗-algebras need not be ambiently unitarily conjugate.
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Invariant projections, hyperinvariant subspaces, and transitive algebras
Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II1 factor.
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Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor .
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The Kadison–Ringrose cohomology conjecture
Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture.
Formal proof listed
Topology2
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The Hilbert–Smith conjecture in every dimension
Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert–Smith conjecture in all finite dimensions, for Hausdorff second-countable manifolds without boundary.
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Four-dimensional disk embedding and Wall's conjecture
The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F2 is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.
Claimed, not checked
Functional analysis1
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Nonexpansive fixed points in reflexive Banach spaces
Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity.
Formal proof listed
Differential geometry5
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Gromov’s integral scalar-curvature bound for simplicial volume
Proves for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with depending only on dimension. Here and is real simplicial volume. Also proves rational inessentiality under positive scalar curvature, resolving the Gromov–Lawson conjecture; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.
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Yau's uniformization conjecture
Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂn.
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Katok's entropy rigidity conjecture
Proves Katok's entropy rigidity conjecture for closed connected Riemannian manifolds of dimension at least three with strictly negative sectional curvature: normalized Liouville measure maximizes entropy for the geodesic flow if and only if the metric is locally symmetric.
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A counterexample to the nearby Lagrangian conjecture
Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.
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The metric Blaschke conjecture
Proves the metric Blaschke conjecture: every closed connected Riemannian manifold of positive dimension whose injectivity radius equals its diameter is, up to scaling, a standard compact rank-one symmetric space.
Claimed, not checked
Partial differential equations1
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De Giorgi's conjecture in dimension eight
Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution of that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions as constant wells or planar transitions, without an energy-growth assumption.
Claimed, not checked