All 372 results

Proves Milne's rationality conjecture for abelian varieties over $\overline{\mathbb Q}$ 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.

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.

Proves that every Dirichlet L-function, including $\zeta(s)$, is zero-free in $\Re s\gt 7/8$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over $\mathbb Q(\sqrt{-3})$. A companion gives a different proof of the zero-free half-plane $\Re s\gt 11/12$.

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.

Proves that Catalan's constant $G=\sum_{j\ge0}(-1)^j/(2j+1)^2$ is irrational.

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.

Proves the ordinary two-point Chowla conjecture, with a bound $O(X/(\log X)^c)$ 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 $n^{it}$ for $|t|\le X$.

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.

Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $K^{\mathrm M}_1/\ell$, $K^{\mathrm M}_2/\ell$, 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.

Every continuous odd absolutely irreducible two-dimensional 2-adic representation of $G_{\mathbb Q}$ 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.

Proves that the normalized ordered logarithms of the prime factors of $p-1$, counted with multiplicity, converge jointly to the Poisson–Dirichlet law $\mathrm{PD}(1)$ as p ranges uniformly over primes up to x and $x\to\infty$. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than $n^{1-\varepsilon}$ totient preimages, for every ε > 0.

Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and $n+1$ are asymptotically independent in ordinary natural density. In particular, the integers satisfying $P^+(n)\lt P^+(n+1)$ have density 1/2.

Proves that no finite modification of the primes can be written as $A+B$ with $A,B\subseteq\mathbb Z_{\ge0}$ each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.

Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over $\overline{\mathbb F}_q$ under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and $p\nmid |W_G|$. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.

Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity.

Proves the abelian Zilber–Pink conjecture over $\overline{\mathbb Q}$: 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 $\mathcal A_2$ for Hodge-generic curves defined over $\overline{\mathbb Q}$, without boundary or reduction assumptions.

Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational $p/q$ satisfies $|\pi-p/q|\ge q^{-2-\varepsilon}$. This also proves convergence of the Flint–Hills series $\sum_{n\ge1}1/(n^3\sin^2 n)$, with angles in radians.

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 $G(k)$ is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.

Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚ_p 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 $X_0(N)$ and $X_1(N)$ of genus at least two.

Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the $(d-2)$-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.

Answers Jacobsthal's quadratic-bound question: every interval of $Ck^2$ consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.

Proves that for every real shift γ and finite-valued $\psi:\mathbb N\to[0,\infty)$, divergence of $\sum_q\phi(q)\psi(q)/q$ implies $\|qx-\gamma\|\lt \psi(q)$ 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.

Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order $X^{5/6}/\log X$. Every fixed nonzero prime-angle Fourier mode has smaller order.

Gives an asymptotic equivalent for the number $V(x)$ of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, $V(cx)/V(x)\to c$ for every fixed c > 0, answering Erdős and Hall’s scaling question.

Every rational $a/b$ with $1\le a\lt b$ is a sum of $O(\log\log b)$ distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.

For every fixed C > 0, a positive proportion of consecutive prime gaps exceed $C\log p_n$, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where $p_n/n$ increases have positive lower density, answering Erdős and Prachar.

Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SL_r character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.

Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound D, the graph joining Gaussian primes at distance at most D has uniformly bounded finite component sizes, depending only on D, including primes on the coordinate axes.

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 $c_a x/(\log x)^2$ primes in every sufficiently large interval $(x,2x)$ have primitive root a, with $c_a\gt 0$.

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.

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.

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. Companion results prove rational Hodge for arbitrary products of projective complex K3 surfaces and algebraicity of the Kuga–Satake correspondence for every such surface.

Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-$\mathcal C$ manifolds with rational simple-normal-crossing boundaries. For projective fibrations $f:U\to V$ of smooth complex quasi-projective varieties with connected fibers, general fiber F, and $\bar\kappa(V)\ge0$, proves Popa's inequality $\bar\kappa(U)\ge\kappa(F)+\max\{\bar\kappa(V),\mathop{\mathrm{Var}}\nolimits (f)\}$, where variation measures the whole geometric generic fiber.

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.

Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p > 3 when the effective boundary is rational and the ℚ-Cartier adjoint is nef of numerical dimension one. The adjoint is semiample, without requiring the original variety to be terminal or ℚ-factorial.

Proves numerical semiampleness for nef adjoints $K_X+B+M$ with $K_X+B$ pseudo-effective and M nef rational, for projective klt rational pairs over algebraically closed characteristic-zero fields and smooth compact Kähler rational klt simple-normal-crossing pairs, using Bott–Chern cohomology in the latter case. Separately, projective generalized log canonical rational pairs over such fields admit minimal models for pseudo-effective adjoints and Mori fiber spaces otherwise, with nef b-data fixed.

Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most $2(n-1)^n$. Equality holds precisely for an analytic ordinary double point.

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 $K_X+mL$ is globally generated for every integer $m\ge n+1$.

Proves Nagata's strict inequality $\sum_i m_i\lt d\sqrt r$ for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities m_i. It also proves maximal multipoint Seshadri constants $(L^n/r)^{1/n}$ 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.

Proves Bloch's conjecture: for every smooth connected projective complex surface S with $p_g(S)=0$, the Albanese map $\mathrm{CH}_0(S)^0\to\mathrm{Alb}(S)(\mathbb C)$ on integral degree-zero zero-cycles is an isomorphism. This combines the new $p_g=q=0$ theorem with the classical theorem of Bloch, Kas, and Lieberman.

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.

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.

Proves $P_k=W_{2k}=W_{2k+1}$ on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant P = W conjecture in every coprime rank.

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.

Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.

Constructs an integral complex affine fourfold $X\not\cong\mathbb A^4$ with $X\times\mathbb A^1\cong\mathbb A^5$, disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections $X\to\mathbb A^1$ and $\mathbb A^5\to\mathbb A^2$ have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.

Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.

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.

Constructs ample rank-two bundles on $\mathbb P^1\times\mathbb P^1$ with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.

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.

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.

Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.

For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3 association. This disproves Kuznetsov's rationality conjecture and the sufficiency of the associated-K3 criterion for rationality; the discriminant threshold is ineffective.

Proves Toda's Gepner conjecture for every smooth complex quintic threefold, constructing a numerical Bridgeland stability condition with the prescribed phase shift 2/5. Also constructs numerical Bridgeland stability conditions at every sufficiently large volume on all smooth projective complex threefolds with trivial canonical bundle, with the exact ordinary and square-root-Todd central charges.

Proves termination of every existing generalized log canonical flip sequence on globally Weil ℚ-factorial compact Kähler fourfolds, with rational boundary, fixed rational analytically nef b-data, and projective small flip diagrams with the prescribed ample signs. Also proves termination of arbitrary permitted minimal model programs for projective log canonical fourfolds with rational boundary in characteristic zero.

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.

Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products $D\times\mathbb C^m\times F$, with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂⁿ has a finite étale cover by an abelian variety.

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.

Every connected minimal compact complex surface of class VII with $b_2\gt 0$ 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 $\mathbb C^2\setminus\{0\}$ whose complement is connected.

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 $b_2=1$ are adjoint varieties with their canonical contact structures, while those with $b_2\ge2$ have underlying manifold $\mathbb P(T^*Z)$ for a smooth projective variety Z.

Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies $\rho(\iota-1)\le n$, with equality exactly for $(\mathbb P^{\iota-1})^\rho$. Here the pseudoindex is the least anticanonical degree of a rational curve.

Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ³ with constant Milnor number, resolving the surface case of the μ-constant problem. After shrinking the parameter disk and representatives, ambient homeomorphisms vary jointly continuously, fix the origin section and preserve the defining functions.

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.

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.

Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.

If X is a smooth connected complex projective variety and $-K_X$ admits a smooth Hermitian metric with nonnegative curvature, then $H^0(X,-mK_X)\ne0$ for some m > 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.

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 $c\in\mathbb C\setminus\mathbb Q$. It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.

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.

Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, $|\phi'|^s$ is area-integrable for $4/3\lt s\lt 4$. The sharp universal integral-means identity is $B_{\mathcal S}(t)=|t|-1$ for t ≤ −2. A strict bound $B_b(-1)\lt 1/4$ for bounded univalent functions disproves Kraetzer's prediction at that parameter.

Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set $E\subset\mathbb R^d$ with Hausdorff dimension greater than $d/2$ determines a set of Euclidean distances of positive Lebesgue measure.

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 $L^3(\mathbb R^3)$ to $L^3(S^2)$ with norm $O_\varepsilon(\delta^{-\varepsilon})$ for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.

Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in $L\log L(\mathbb T)$ converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the $L\log L$ scale.

Constructs polynomials with N consecutive coefficients in $\{-1,1\}$ whose modulus is $(1+o(1))\sqrt N$ uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.

Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface $\Sigma\subset\mathbb R^3$, including surfaces with boundary, the extension operator is bounded from $L^p(\Sigma)$ to $L^p(\mathbb R^3)$ for every p > 3.

Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on $L^p(\mathbb R^3)$ for every $1\le p\le\infty$ whenever $\delta\gt \max\{3|1/p-1/2|-1/2,0\}$.

Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range $2\lt p\lt \infty$, with Sobolev regularity above $\max\{0,1-3/p\}$. In particular, the critical L³ estimate holds with every positive Sobolev loss.

Proves almost-everywhere convergence $e^{it\Delta}f\to f$ as $t\downarrow0$ for every $f\in H^{n/(2(n+1))}(\mathbb R^n)$ and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H^1/3.

Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and $2\le n\le d-2$, an n-Ahlfors–David regular Radon measure on ℝ^d 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.

Proves maximal and annular r-variation bounds, for every r > 2, from complex $L^3(\mathbb R^2)\times L^3(\mathbb R^2)$ to $L^{3/2}(\mathbb R^2)$. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $L^3\times L^3\times L^3$ point.

Proves a uniform strong L² bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an L²-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale.

For every fixed $q\in(0,1)$, constructs compact subsets of $[0,1]$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of $\{q^n:n\ge1\}$, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.

Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real $f\in W^{1,1}(\mathbb R^2)$, the centered disk maximal function satisfies $\|\nabla Mf\|_1\le C\|\nabla f\|_1$ with an absolute constant. It belongs locally to $W^{1,1}$ and has a globally integrable weak gradient.

Proves that the principal-value trilinear Hilbert transform with shifts $x-t$, $x-2t$, $x-3t$ is bounded from $L^3(\mathbb R)^3$ to $L^1(\mathbb R)$. This resolves the L³ exponent case of the standard conjecture for slopes 1, 2, 3.

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 $K\times K^\circ$ in dimension $2n$ has Gromov width 4.

Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.

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.

Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for $0\lt s\lt 2$ and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.

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 L_p Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout $0\lt p\lt 1$.

Determines the optimal worst-case covering density as $\Theta(n\log n)$, for both lattice and unrestricted translative coverings. Every convex body in ℝⁿ, n ≥ 2, admits a lattice covering of density at most $Cn\log n$; centrally symmetric examples in every sufficiently large dimension require at least $cn\log n$ even without the lattice restriction, for absolute $c,C\gt 0$.

Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian parameter.

For every fixed $1\lt p\lt \infty$ and distortion D > 1, every n-point subset of real L_p embeds into $\ell_p^d$ with distortion at most D and dimension $d=n^{o(1)}$, answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension $\Theta(n^2)$ when p ≠ 2.

Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.

Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most $9/(8\pi)$. In dimension at least two, three planar sectors of angle $2\pi/3$, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor $(8\pi/9)(1-1/k)$ for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.

Proves that any finite sequence in the Euclidean unit ball of ℝ^d admits signs keeping every partial sum within $C\sqrt d$, independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order $S_2(d)=\Theta(\sqrt d)$.

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.

Determines the least distortion of embedding edit distance on words of length at most d into real ℓ₁: it is $\exp(\Theta(\sqrt{\log d\,\log\log d}))$. 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.

Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.

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.

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.

Proves $\mathsf L=\mathsf{RL}=\mathsf{BPL}$, 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.

Gives deterministic algorithms using $2^{O((\log(L+2))^2)}$ 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.

Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational $\delta\in(0,1)$, 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.

It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed $0\lt \delta\lt 1/3$, 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 $\delta n$ otherwise, where n is the number of vertices.

Proves $\omega\le9/4$ over ℂ, giving $O_\varepsilon(n^{9/4+\varepsilon})$ arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension n^a with a > 0.465 permits $n^{2+o(1)}$ rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.

Proves an $\Omega(n^3)$ lower bound for the border determinantal complexity of the $n\times n$ permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least $cn^3$, for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations.

Multiplies two n-bit integers exactly at every input length in deterministic worst-case time $O(n(\log n)^{1-\kappa})$, with $\kappa=2^{-182}$, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen $n\log n$ optimality conjecture in the ordinary multitape bit model.

Establishes a randomized competitive ratio $O(\log^2(k+1))$ for k-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on infinite unbounded metrics. On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and $\log(t+1)$ at request t, with a finite instance-dependent additive movement constant.

For every finite matroid known in advance, gives a distribution-independent online rule using one independent sample per element and earning a universal constant fraction of the expected offline optimum. Values are independent and nonnegative, with finite expected optimum. The guarantee holds even when the arrival-order adversary sees all samples, values, and the rule's entire random seed; no polynomial-time implementation is asserted.

Constructs a single language in deterministic polynomial time whose n-bit membership function requires $2^{\omega(\sqrt n)}$ total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds.

Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points.

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.

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.

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.

Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least $\sqrt{\log n}/(\log\log n)^3$, approaching the square-root-logarithmic upper bound.

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.

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.

Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in $(n+m)^{1+o(1)}$ word time, with success probability at least 2/3. The time bound holds on every computation path. The same guarantees apply to finding a spanning subgraph with prescribed admissible vertex degrees, or deciding that none exists.

For every fixed rational $\varepsilon\in(0,1)$, gives a randomized $(1+\varepsilon)$ approximation to unit-cost edit distance in worst-case expected time $N^{1+o(1)}$, with success probability at least 2/3. The strings have total length N and polynomially bounded integer symbols. This is an asymptotic guarantee at fixed accuracy.

At every fixed deletion probability in $(0,1)$, reconstructing an arbitrary length-n binary string requires $n^{\Omega(\log\log n)}$ 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 $n^{-\varepsilon}$ for fixed ε > 0, both bounds become polynomial in the input and parameter encoding.

Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule.

Gives a deterministic polynomial-time $(1+2/e+\varepsilon)$-approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming $P\ne NP$, the optimal infimum approximation factor is $1+2/e$.

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 $0\lt \rho\lt 1$, allowing arbitrary real positive semidefinite factors.

Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most $8d\sqrt n$, uniformly for $1\le d\le n$. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with $\mathop{\mathrm{sign}}\nolimits (0)=1$.

Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels.

Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda–Sipser state-succinctness conjecture over growing finite alphabets; both results rule out polynomial state bounds independent of alphabet size.

Gives a deterministic length-n discrete Fourier transform algorithm using $O(n(\log n)^{1-\delta})$ operations for every n, with explicit $\delta=10^{-13}$. The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.

Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in $O(n^8)$ 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.

Constructs total Boolean functions with block sensitivity $\mathop{\mathrm{bs}}\nolimits (f)\ge s(f)^\alpha$ for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here $s(f)$ counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together.

Proves unconditional $n^{\Omega(k)}$ deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete.

Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem.

Over every characteristic-zero field, the $(1,1)$ entry of a product of n independent $n\times n$ variable matrices requires $n^{\Theta(\sqrt n)}$ gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.

Resolves the quasilinear PCP-for-PPAD conjecture. An End-of-Line instance of length N reduces to numerical circuit constraints of total length $N(\log N)^{O(1)}$ such that any polynomially encoded rational assignment satisfying all but a fixed fraction to fixed accuracy yields an endpoint solution. Such assignments always exist, giving robust local verification with only quasilinear size overhead.

Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using $O(T^{2/5}\log^C(T+2))$ space, improving the square-root exponent. Heads move at most one cell per step; finitely many read-only input heads are allowed. Initial contents are independent of T, and contents and input symbols have polylogarithmic-space access. Simulation time is unrestricted.

Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary $O(2^{0.49n})$ word-RAM time on polynomial-bit inputs, where n counts the integers. The time bound holds on every execution and success probability is at least 2/3 on every input. Inputs may repeat positive integers; words have $O(n+b)$ bits for maximum input bit length b.

For C² potentials with a supplied minimizer and $I\preceq\nabla^2V\preceq2I$, proves that sampling within total variation 1/10 requires only $C_\varepsilon d^\varepsilon$ exact value-and-gradient queries for every fixed ε > 0. The bound holds on every run, with unrestricted computation between queries. A logarithmic lower bound also holds, so the optimal power-law exponent in this oracle model is zero.

For fixed A > 0, a one-pass learner with $Ad^2$ persistent bits needs $\Omega_A(d\log(1/\epsilon))$ noiseless Gaussian samples to recover a unit vector to angular error $0\lt \epsilon\le1/10$ with probability 2/3, uniformly in accuracy for large d. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness.

Proves that the existential theory of the reals lies in the counting hierarchy. More generally, truth of existential–universal real sentences can be decided at one fixed level of that hierarchy, even when their integer polynomials are specified by arithmetic circuits.

Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽_p, including multiplicities, in bit complexity polynomial in $(n+1)\log p$. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.

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.

Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L² space: the bilateral iterates of one real observable form an orthonormal basis of that space.

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.

Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail.

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.

For every self-similar measure on the line generated by finitely many contracting similarities, proves $\dim_{\mathrm H}\mu=\min\{1,h_{\mathrm{RW}}/\chi\}$, where $h_{\mathrm{RW}}$ is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios.

Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set. All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds.

Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions.

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.

Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C^∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.

Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every $\lambda\in(0,1)$ by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in $(1,2)$, giving examples beyond reciprocal Pisot parameters.

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.

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.

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.

Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy $\chi_f(G)\gt h(G)$, where $h(G)$ is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound $\chi_f(G)\le\mu(G)+1$. In the positive direction, every finite nonempty graph satisfies $\chi_{\mathrm{list}}(G)\le C h(G)$ for a universal constant C.

Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane.

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 $\{1,\ldots,N\}$ with no nonconstant k-term progression has size at most $C_kN\exp[-c_k(\log N)^{\varepsilon_k}]$, with positive constants depending only on k.

Resolves Erdős's superexponential-growth question for van der Waerden numbers. If $W_r(k)$ is the least interval length forcing a monochromatic k-term progression in every r-coloring, then $W_r(k)\gt k^{ck\lfloor\log_2 r\rfloor}$ for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, $W_r(k)^{1/k}\to\infty$ for each fixed r.

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.

Disproves Ryser's covering conjecture by constructing intersecting $(q+1)$-partite, $(q+1)$-uniform hypergraphs with covering number $q+1$, 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.

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.

Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs.

For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝ^d determine at least $c_dn^{2/d}$ distinct distances, with $c_d\gt 0$ depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.

Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but $o(n)$ points of any n-point planar set determine at least $n^{1-\varepsilon}$ distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by $O(n^{4/3-\delta})$ for an absolute δ > 0.

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.

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 $\mathbb N[q]$. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity.

For every fixed integer s ≥ 5, proves $r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)}$ as $t\to\infty$, determining the logarithmic exponent and matching the classical upper bound at that scale. Here $r(s,t)$ is the least number of vertices forcing an s-clique or a t-vertex independent set.

Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the n-dimensional cube is $\Theta(2^n)$. 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.

Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set.

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.

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 $C/k$ fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities.

Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices.

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 $G(n,p)$ is at most $C p_{\mathrm E}(n,H)(1+\log_2 h)$, with universal C. Here $p_{\mathrm E}$ is the least density at which every subgraph of H has expected copy count at least 1/2.

Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽₂ coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of such expanders in every positive dimension.

For every fixed d ≥ 3, constructs a simple d-regular nonbipartite Ramanujan graph on every sufficiently large even number n of vertices, with every nonconstant adjacency eigenvalue strictly between $-2\sqrt{d-1}$ and $2\sqrt{d-1}$. A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on d.

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 $n\in\{2,3,4,5,7,11,13\}$.

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.

Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most $Cn$ 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.

For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of $\{1,\ldots,N\}$ avoiding nonzero values $h(1),h(2),\ldots$ as differences has size $O_h(N^{1-c_k})$, with $c_k\gt 0$ depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h.

Improves the planar halving-line bound to $O(n^{4/3-\varepsilon})$ for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has $O(n(k+1)^{1/3-\varepsilon_0})$ strictly separable k-subsets for $1\le k\le n/2$, with an absolute $\varepsilon_0\gt 0$. The constants and positive exponents are nonquantitative.

Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need $O_F(\Delta/\log\Delta)$ colors when their maximum degree Δ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed r ≥ 4, every n-vertex K_r-free graph of average degree d ≥ 2 has an independent set of size $\Omega_r(n\log d/d)$.

Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result.

Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed $0\lt \varepsilon\lt 1/2$, every nontrivial increasing relabeling-invariant property crosses from probability ε to $1-\varepsilon$ within width $O((\log n)^{-2})$ for graphs and $O_r((\log n)^{-r/(r-1)})$ for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties.

Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins.

Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to $n^{3/2}/(2\sqrt2)$, with mean-square convergence after normalization by n^3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture.

Proves the Erdős–Faudree–Rousseau–Schelp conjecture: $R(C_m,K_n)=(m-1)(n-1)+1$ for every $m\ge n\ge3$, except $R(C_3,K_3)=6$. This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph.

Disproves polynomial ordered binary matrix removal with one fixed $66\times66$ zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance. Copies preserve row and column orders and match both zeros and ones.

For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least $n^{-2+c}$ for one absolute c > 0. This disproves the conjectured almost-n⁻² upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be.

Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with $\sum_i\widehat f(\{i\})\gt C\sqrt{\deg(f)}$. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.

Proves strict positivity of Serre's intersection multiplicity $\chi^R(M,N)$ for nonzero finitely generated modules over any regular local ring, provided $M\otimes_R N$ has finite length and $\dim M+\dim N=\dim R$. This resolves the positivity conjecture, including ramified mixed characteristic.

Proves $e(R)\le e(S)$ 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.

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.

Constructs a finitely presented torsion-free group G whose group algebra $\mathbb F_2[G]$ has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.

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.

Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.

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.

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.

Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem on local finiteness negatively.

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.

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.

Proves saturation factor one for $\mathop{\mathrm{Spin}}\nolimits (2n)$, 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.

Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every S_n with $n\notin\{2,4,9\}$ has an irreducible representation whose tensor square contains all irreducibles.

Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group.

Proves the ℓ¹-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over $\ell^1(G)$ 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.

Proves that every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal fiber functor to a higher Verlinde category $\mathrm{Ver}_{p^n}$. The level may depend on the category, and the theorem includes characteristic two, resolving the finite case of the Benson–Etingof–Ostrik conjecture.

Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic $(0,5)$ have nonzero integral K-theory classes that vanish over their fraction fields.

Proves the sixth case of Foulkes’ conjecture: $\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^bV)$ embeds equivariantly in $\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6V)$ for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map $\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^aV)\to\mathop{\mathrm{Sym}}\nolimits ^a(\mathop{\mathrm{Sym}}\nolimits ^bV)$ is surjective for a ≥ 2 and $b\ge a(a-1)$, giving dimension-independent quadratic stabilization.

Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for $0\lt q\le4$ and to the Brownian continuum random tree for q > 4, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs.

Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C¹ boundary. Differentiability also holds for every Gamma law with positive shape and rate.

Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with $p_c\lt 1$: 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.

Proves $p_c\lt p_u$ 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.

Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $O(3)$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $O(4)$ model, determines the exact leading asymptotic of the full transfer gap, $m_{\mathrm{lat}}(\beta)\sim32e^{\pi/4-1/2}\sqrt\beta\,e^{-\pi\beta}$. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $O(n)$ models with n ≥ 3 at every positive temperature.

For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations $C_{\beta_c}(r)\sim Ar^{-1/4}(\log r)^{1/8}$ and the Berezinskii–Kosterlitz–Thouless essential singularity $\sqrt{\beta_c-\beta}\log\xi(\beta)\to B$, with $A,B\gt 0$ after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.

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 $c_\beta n^{1/3}$, with $c_\beta\gt 0$, confirming the predicted n^1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.