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Paper The optimal order of convex-body covering density
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…
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Paper Finite Smith–Toda complexes at every height
For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology BP/(p,v1,\ldots,vn) and the canonical comodule structure. Thus Smith–Toda complexes exist at every…
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Paper The Euclidean plane cannot be colored with five colors
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…
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Paper Bounded-distortion L_{1} embeddings of planar and bounded-treewidth graphs
Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L1 with universal…
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Paper Yau’s nodal bounds: surfaces and higher dimensions
Proves the sharp C\sqrt\lambda upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four,…
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Paper The L\log L Fourier-convergence conjecture
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…
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Paper The Falconer distance conjecture
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…
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Paper Graph coloring, clique minors, and Colin de Verdière invariants
Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy \chif(G)\gt h(G), where h(G) is the largest clique-minor order.…
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Paper 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…
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Paper A counterexample to periodic tiling in dimension three
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…
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Paper Rokhlin’s multiple-mixing problem
Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection…
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Paper Banach’s simple Lebesgue-spectrum problem
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…
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Paper Tingley’s sphere-isometry problem
Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is…
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Paper Independence of the separable quotient problem
Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable…
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Paper Thompson's group F is nonamenable
Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem. () Manuscript September 23, 2026 · · Lean…
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Paper 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…
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Paper 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 S^{m+1} with non-totally-geodesic image has…
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Paper Tangent splittings and product decompositions
A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville's…
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Paper 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…
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Paper Positive metric entropy for the standard map
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…
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Paper 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…
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Paper Gaussian free field limits throughout the balanced six-vertex regime
The balanced square-lattice six-vertex height field with a=b=1 and 0\lt c\le2 converges to a Gaussian free field, including at the endpoint c = 2. The plane state is defined by balanced-torus limits. For unit…
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Paper Approximate counting and entropy of perfect matchings
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…
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Paper Markov type characterizes superreflexivity
Proves that every real Banach space with Markov type p for some p > 1 admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes…
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Paper Beyond the square-root exponent for depth-three circuits
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…
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Paper One-sample matroid prophet inequalities against an almighty adversary
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…
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Paper 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 T^(S^9\times S^{N-1}) that is diffeomorphic to the base but not…
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Paper Integer multiplication below n\log n
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…
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Paper 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…
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Paper The Mézard–Parisi formula for diluted spin glasses
Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The…
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