Papers

For every fixed $k\ge2$, a deterministic algorithm counts the satisfying assignments of a $k$-CNF with $n$ variables and $m$ clauses in time $(n+m)^{O_k(1)}2^{n/(\log n)^{1/2+o(1)}}$. This yields superlinear circuit lower bounds for a language in $\mathsf{E}^{\mathsf{NP}}$ on infinitely many input lengths.

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.

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.

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.

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.

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 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.

Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽$_{2}$ 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.

Resolves the free group factor isomorphism problem: $L(\mathbb F_2)\cong L(\mathbb F_3)$, and hence all interpolated free group factors, including $L(\mathbb F_\infty)$, are isomorphic. Their common factor has fundamental group $\mathbb R_{\gt 0}$.

Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval.

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 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 $|t|\le X$.

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 the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability.

Proves Bloch's T$^{3/2}$ law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ$^{3}$. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings.

Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂ$^{n}$.

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.

Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in $T^*(S^9\times S^{N-1})$ that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.

The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin $K(\pi,1)$ conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT$(0)$ space.

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.

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.

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.

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.

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 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.

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.

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.

Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II$_{1}$ factor.

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 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 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 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.