Named conjectures
Results whose title names a conjecture: a statement mathematicians believed but could not prove.
85 AI-made results on named conjectures, from 120 papers in the OpenAI math collection, each explained in plain English. 43 have a formal Lean proof listed; 42 are claimed in a paper and not yet machine-checked. They span 16 fields, led by algebraic and complex geometry (12), combinatorics (12) and algebra (11). The oldest question here dates back to 1902: The Hilbert–Smith conjecture in every dimension. Results rated Landmark include The rational Hodge conjecture for CM abelian varieties and Nagata’s conjecture and maximal Seshadri constants.
Number theory9
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Milne’s rationality conjecture and algebraic specialization
Proves Milne's rationality conjecture for abelian varieties over with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.
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Goldfeld’s conjecture: densities and mean analytic rank
Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.
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Ordinary two-point correlations and the corrected Elliott conjecture
Proves the ordinary two-point Chowla conjecture, with a bound for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times for .
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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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Ostmann’s inverse Goldbach conjecture
Proves that no finite modification of the primes can be written as with each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.
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The Margulis–Platonov conjecture over global fields
Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract normal subgroup of is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.
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The local p-adic section conjecture and global consequences
Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚp correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over ℚ for the modular curves and of genus at least two.
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The weak inhomogeneous Duffin–Schaeffer conjecture
Proves that for every real shift γ and finite-valued , divergence of implies for infinitely many q, for almost every x. Here ϕ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.
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Uchida’s conjecture for open homomorphisms of Galois groups
Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the opposite direction. No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.
Claimed, not checked
Algebraic and complex geometry12
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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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Fujita’s freeness conjecture
Proves Fujita's freeness conjecture at its sharp bound in every dimension: for a smooth projective complex variety X of dimension n and an ample line bundle L, the adjoint is globally generated for every integer .
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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.
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Bloch’s conjecture for complex surfaces
Proves Bloch's conjecture: for every smooth connected projective complex surface S with , the Albanese map on integral degree-zero zero-cycles is an isomorphism. This combines the new theorem with the classical theorem of Bloch, Kas, and Lieberman.
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The equivariant cohomological Hikita conjecture
Proves the equivariant cohomological Hikita correspondence for every finite quiver, including loops and multiple arrows, with arbitrary dimension and framing vectors and commuting flavor torus. When every semistable point is stable and the gauge action is free, the equivariant cohomology of the Nakajima variety is canonically the coordinate ring of the scheme-theoretic cocharacter-fixed locus of its flavor-deformed Coulomb branch.
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A counterexample to Griffiths’ positivity conjecture
Constructs ample rank-two bundles on with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.
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Kobayashi’s canonical-ampleness conjecture
Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler setting.
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Counterexamples to Zariski’s multiplicity conjecture
Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ4 with isolated critical points and multiplicities four and five.
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The Global Spherical Shell conjecture
Every connected minimal compact complex surface of class VII with contains a global spherical shell, proving the positive-b2 Global Spherical Shell conjecture. Such a shell is a holomorphically embedded neighborhood of the standard three-sphere in whose complement is connected.
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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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The generalized Mukai conjecture
Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies , with equality exactly for . Here the pseudoindex is the least anticanonical degree of a rational curve.
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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.
Claimed, not checked
Real and complex analysis6
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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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Brennan's conjecture and the integral-means spectrum
Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, is area-integrable for . The sharp universal integral-means identity is for t ≤ −2. A strict bound for bounded univalent functions disproves Kraetzer's prediction at that parameter.
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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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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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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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The geometric case of the Erdős similarity conjecture
For every fixed , constructs compact subsets of with measure arbitrarily close to one containing no translated and nontrivially dilated copy of , with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.
Formal proof listed
Convex and metric geometry4
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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.
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Logarithmic and Lp Brunn–Minkowski inequalities and the B-conjecture
Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout .
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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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The sharp simplex conjecture for isotropic constants
Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.
Claimed, not checked
Theoretical computer science2
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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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The Courtade–Kumar and Hellinger conjectures
Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation.
Formal proof listed
Dynamical systems and ergodic theory2
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The near-boundary Birkhoff conjecture
Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.
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A C1 counterexample to the entropy conjecture
Constructs a noninvertible C1 self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C1 self-maps: homological growth need not force positive orbit complexity.
Formal proof listed
Combinatorics12
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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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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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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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Counterexamples to Ryser’s covering conjecture
Disproves Ryser's covering conjecture by constructing intersecting -partite, -uniform hypergraphs with covering number , rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture.
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Hindman’s finite sums and products conjecture
Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.
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The higher-dimensional Erdős distinct-distances conjecture
For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least distinct distances, with depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.
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The hypercube Ramsey conjecture
Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the n-dimensional cube is . Thus every red-blue coloring of a complete graph on a universal constant times the cube's number of vertices contains a monochromatic copy of the cube.
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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.
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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.
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The circulant Hadamard and Barker-sequence conjectures
Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when .
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Barnette’s Hamiltonian-cycle conjecture
Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle.
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The Erdős–Gallai cycle-decomposition conjecture
Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs.
Formal proof listed
Algebra11
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Serre’s intersection-multiplicity conjecture
Proves strict positivity of Serre's intersection multiplicity for nonzero finitely generated modules over any regular local ring, provided has finite length and . This resolves the positivity conjecture, including ramified mixed characteristic.
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Lech’s multiplicity conjecture
Proves for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic.
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A counterexample to the small Cohen–Macaulay module conjecture
Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over ℂ has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture.
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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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Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures
Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field.
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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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Saxl’s conjecture and universal tensor squares
Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every Sn with has an irreducible representation whose tensor square contains all irreducibles.
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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.
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Integral counterexamples to Gersten’s conjecture
Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic have nonzero integral K-theory classes that vanish over their fraction fields.
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Foulkes' conjecture for sixth powers and quadratic stabilization
Proves the sixth case of Foulkes’ conjecture: embeds equivariantly in for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map is surjective for a ≥ 2 and , giving dimension-independent quadratic stabilization.
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Probability and statistical mechanics1
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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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Group theory2
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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.
Claimed, not checked
Mathematical physics1
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The ionization and generalized ionization conjectures
For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing m electrons has Thomas–Fermi asymptotics as and ; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking first.
Formal proof listed
Operator algebras3
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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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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
Topology7
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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.
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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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Quillen's conjecture in rational homology
Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.
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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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The four-dimensional Singer conjecture
Proves that the L2-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.
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Curtis’s conjecture
Proves Curtis’s conjecture: the positive-degree mod-two stable Hurewicz image of the sphere is spanned by the images of the Hopf-invariant-one classes η, ν, σ and the Kervaire-invariant-one classes that exist.
Claimed, not checked
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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The cotype–cotype conjecture under the approximation property
Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type.
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Differential geometry7
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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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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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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.
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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.
Claimed, not checked
Partial differential equations4
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The planar Mumford–Shah regularity conjecture and local weak-L4 gradient bounds
Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a arc, a regular crack tip, or three arcs meeting at ; only finitely many global connected components meet any compact interior region.
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The hot spots conjecture for simply connected planar domains
Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.
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The Lane–Emden and Hénon–Lane–Emden conjectures
Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, and real A, B, the system , has no positive entire solution when , with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and .
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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