Open problems from the 2000s
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. Results rated Landmark include The Unique Games Conjecture and optimal approximation thresholds and Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling.
Number theory2
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Irrationality of Catalan’s constant
Proves that Catalan's constant is irrational.
Formal proof listed -
Zilber–Pink in abelian varieties and the Siegel threefold
Proves the abelian Zilber–Pink conjecture over : every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold for Hodge-generic curves defined over , without boundary or reduction assumptions.
Claimed, not checked
Algebraic and complex geometry6
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Hyperkähler SYZ and projective-space bases
Proves the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. It also proves that every projective Lagrangian fibration with normal projective base has projective space as its base, in every dimension and deformation type.
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Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII
Proves that every complex K3 surface is Oka, including nonprojective surfaces. More generally, every connected minimal compact complex surface of Kodaira dimension zero is Oka; a connected minimal compact complex surface of class VII is Oka exactly when it is a Hopf or Enoki surface.
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A stable-coordinate counterexample in four variables
Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces.
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Tangent splittings and product decompositions
A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville's splitting conjecture. On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring's conjecture and the corresponding product decomposition.
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Fundamental groups of special complex varieties and root orbifolds
Proves Campana's abelianity conjecture: special compact Kähler manifolds have virtually abelian fundamental groups. Using this theorem, establishes the same conclusion for order-two root orbifolds of smooth projective complex fourfolds along one nonempty smooth connected divisor, when special in the stated differential-line sense. For smooth special complex quasi-projective varieties, proves that every finite-dimensional complex linear representation of the fundamental group has virtually nilpotent image of class at most two.
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Global quantum geometric Langlands at irrational level
Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level . It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.
Claimed, not checked
Real and complex analysis1
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Endpoint Sobolev regularity of centered disk averages
Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real , the centered disk maximal function satisfies with an absolute constant. It belongs locally to and has a globally integrable weak gradient.
Formal proof listed
Convex and metric geometry4
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Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs
Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L1 with universal distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded.
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The Gaussian propeller conjecture in every dimension
Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most . In dimension at least two, three planar sectors of angle , extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.
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Compact counterexamples to bi-Lipschitz dimension reduction
Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
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The sharp exponential scale of edit-distance distortion
Determines the least distortion of embedding edit distance on words of length at most d into real ℓ1: it is . Insertions, deletions and substitutions have unit cost. The constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with d; binary words already force the lower bound.
Formal proof listed
Theoretical computer science8
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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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Perfect completeness for 2-to-1 games
Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational , it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted instances. The alphabet depends only on δ, and every right-hand label has exactly two preimages under each constraint map.
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Hardness of coloring three-colorable graphs
It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed , a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size otherwise, where n is the number of vertices.
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Approximate counting of common integer polymatroid bases
Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ.
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Sampling and counting contingency tables with arbitrary margins
For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution.
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Uniform black-box noncommutative identity testing across characteristics
For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation.
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Quantitative trace-reconstruction bounds with a uniform decoder
At every fixed deletion probability in , reconstructing an arbitrary length-n binary string requires independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most for fixed ε > 0, both bounds become polynomial in the input and parameter encoding.
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The metric k-median approximation threshold and recovery
Gives a deterministic polynomial-time -approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming , the optimal infimum approximation factor is .
Formal proof listed
Combinatorics2
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Deterministic construction of strong thin spanning trees
Resolves the strong thin-tree conjecture constructively. Every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities.
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The second Kahn–Kalai conjecture with an edge-count bound
Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in is at most , with universal C. Here is the least density at which every subgraph of H has expected copy count at least 1/2.
Formal proof listed
Algebra1
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Tensor saturation for even spin groups
Proves saturation factor one for , n ≥ 2: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already gives an invariant at the original weights. This resolves the type-D part of the simply-laced saturation conjecture.
Claimed, not checked
Probability and statistical mechanics6
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The Mézard–Parisi formula for diluted spin glasses
Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability.
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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.
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Exact Hausdorff gauges for SLE
Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, . The explicit gauge , , gives almost surely positive finite measure to every trace segment with , and finite expected measure to the trace in every bounded disk.
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Gaussian fields and interfaces for triangular-lattice Lipschitz heights
Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments and two-arc boundary values , and zero-boundary integer Lipschitz heights weighted by fixed . Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal SLE4, establishing Schramm’s real-field/interface predictions.
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Optimal logarithmic mixing of the Thorp shuffle
Proves that the Thorp shuffle randomizes labeled cards in physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions.
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Sharp singularity rates for symmetric random sign matrices
Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give ; for fixed bias , the rate is . In the biased case, agreeing rows attain this rate.
Claimed, not checked
Mathematical logic1
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Rigidity of the Turing degrees
Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability.
Formal proof listed
Mathematical physics4
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Area laws and tensor networks for two-dimensional gapped systems
Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L−1.
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Entanglement without distillable secret key
Constructs an entangled state on with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on whose square is not entanglement breaking disproves Christandl's PPT-square conjecture.
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The entropy photon-number inequality
Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output's entropy photon number is at least the transmissivity-weighted average of the inputs'. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ.
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Polynomial-time unitary synthesis from a Boolean oracle
Solves the constant-error Aaronson–Kuperberg unitary synthesis problem: a uniform polynomial-size quantum oracle circuit approximates every n-qubit unitary channel within diamond-norm error 1/2, after a suitable Boolean oracle is chosen. Gates, qubits, oracle calls and query length are polynomially bounded. The target-dependent oracle may have an unrestricted truth table; its efficient classical construction is not asserted.
Claimed, not checked
Operator algebras4
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Kirchberg's norm-ultrapower embedding problem
Constructs an explicit separable unital full group C∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C∗-algebra, for any free ultrafilter on the natural numbers. Taking the target to be answers Kirchberg's norm-ultrapower embedding problem negatively.
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A ZFC counterexample to Naimark's problem
Gives an alternative to [Tanaka's ZFC construction](https://arxiv.org/abs/2609.26930v1) of a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable.
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Radius of comparison equals half the mean dimension
For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of equals , including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.
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Weak pure infiniteness and Cuntz-algebra absorption
Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs , without unitality or simplicity assumptions.
Formal proof listed
Functional analysis2
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The complete Crouzeix conjecture
Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has , where is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.
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A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively. Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound.
Formal proof listed
Differential geometry6
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Smooth isometric immersions of surfaces into ℝ4
Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ4, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.
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Donaldson's hypersymplectic deformation conjecture
Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by , deforms through positive closed triples to a hyperkähler triple while preserving all three cohomology classes. Any positive triple can first be normalized by a constant linear change.
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Donaldson's tamed-to-compatible conjecture
Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change.
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Nonnegative-curvature Einstein classification and an L2 topological gap
Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S4, Fubini–Study , or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L2 norm of its trace-free Ricci curvature lies below a universal positive constant.
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The isoperimetric profile of the cubic three-torus
Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are and , with exactly the adjacent two types minimizing at each transition.
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Weak MTW curvature gives convexity and regular optimal transport
On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous.
Formal proof listed
Partial differential equations1
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The three-dimensional Ball–Evans approximation problem
Resolves the three-dimensional Ball–Evans approximation problem: every homeomorphism between arbitrary bounded domains in ℝ3, for , is a strong limit of smooth diffeomorphisms onto the same target.
Claimed, not checked