Algebra
16 recordsGroups, rings, representations, and algebraic structures, including their connections to geometry and topology.
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Broad topics
Algorithms and computation
19 recordsAlgorithms for graphs, strings, arithmetic, and matrix multiplication, together with computational complexity.
Analysis and probability
24 recordsFunctions, operators, differential equations, dynamics, and random systems, including analytic number theory and mathematical physics.
Combinatorics
14 recordsPatterns, colorings, and discrete constructions, including Ramsey theory, cap sets, and geometric combinatorics.
Geometry and topology
30 recordsShapes, distances, spaces, and their structure, including packings, metric embeddings, manifolds, and algebraic geometry.
Logic and formal proof
13 recordsMathematical foundations, decidability, proof systems, and AI-assisted formalization.
Number theory
19 recordsPrime numbers, arithmetic functions, rational approximations, and Diophantine questions.
Optimization and approximation
8 recordsHow close can an efficiently found solution come to the optimum?
Focused topics and problems
AI-assisted formal proof
9 recordsFormal proof systems check deductions expressed in a precise language.
Algebra and representation theory
2 recordsAlgebra studies rings, modules, and categories.
Algebraic and complex geometry
5 recordsPolynomial equations define geometric spaces.
Cap sets
1 recordA cap set is a subset of vectors over the three-element field containing no three distinct vectors in an arithmetic progression.
Convex-body coverings
2 recordsCovering space with translates of convex shapes.
Differential geometry
2 recordsDifferential geometry studies smooth spaces and metrics.
Dynamical systems and ergodic theory
1 recordDynamical systems describe how states evolve.
Functional analysis
1 recordFunctional analysis studies spaces of functions and operators.
Geometric combinatorics
6 recordsColoring, diameter coverings, and packings combine geometric constraints with discrete choices.
Graph and computational algorithms
10 recordsAlgorithms for graph problems, string comparison, arithmetic, and exact search.
Group theory
3 recordsGroups encode symmetry.
Kissing numbers
2 recordsHow many equal, non-overlapping spheres can touch a central sphere?
Mathematical logic and foundations
3 recordsLogic studies formal languages, models, and computability.
Mathematical physics
3 recordsThis topic follows mathematical statements in physical models, including gravitational inequalities and field equations; it does not turn a theorem into an experimental prediction.
Matrix multiplication
4 recordsMultiplying two matrices is a basic computational task.
Metric embeddings
3 recordsHow faithfully can one distance space be represented in another?
Online bin packing
1 recordItems arrive one at a time and must be placed into fixed-capacity bins.
Operator algebras
1 recordOperator algebras organize bounded operators and their symmetries.
Partial differential equations
4 recordsPDE research follows existence, uniqueness, regularity, and blowup for specified equations and initial data.
Probability and statistical mechanics
4 recordsProbability studies random structures and their limits.
Ramsey bounds
2 recordsHow large must a graph be before every coloring of its edges contains a prescribed monochromatic pattern?
Real and complex analysis
4 recordsAnalysis studies functions, limits, and transformations.
Topology
3 recordsTopology studies spaces and continuous transformations.
A Sidorenko counterexample
1 recordSidorenko’s conjecture predicts the lower bound for every bipartite pattern.
A torsion-free direct-finiteness counterexample
1 recordCharacteristic-zero and sofic-group direct-finiteness theorems do not settle arbitrary positive-characteristic group algebras.
AI-assisted counterexample to Grothendieck’s group-scheme question
1 recordAkhil Mathew’s contribution gives a noncommutative finite free group scheme of order four over a nonreduced base that is not killed by four.
AI-assisted realization of the Mathieu group M₂₃ over the rationals
1 recordHuang, Jackson, Lee, Poonen, Pries and Zhang construct an explicit degree-23 polynomial whose splitting field over the rationals has Galois group M₂₃.
AI-assisted small counterexamples to the Gaussian Moments Conjecture
1 recordChristopher D.
Almost-linear approximation of edit distance
1 recordComputing edit distance compares strings under insertion, deletion, and substitution.
AlphaDev: faster small sorting routines in LLVM libc++
1 recordDeepMind reports AI-discovered assembly sorting routines translated to C++ and incorporated in LLVM libc++.
AlphaEvolve-assisted large hypercubes in Bruhat order
1 recordEllenberg, Libedinsky, Plaza, Simental and Williamson prove that for n=2^m, dyadically well-distributed permutations form a Bruhat interval isomorphic to a hypercube of dimension m·2^(m−1).
Alpöge–Buckmaster: Boussinesq blowup with smooth forcing
1 recordAlpöge and Buckmaster report finite-time blowup for the inviscid Boussinesq equations on R² with smooth spatially compact forcing in both equations.
Alpöge–Claude reports: a complex structure on the six-sphere
1 recordThe Alpöge–Claude manuscript claims a compact complex threefold, obtained by completing a modular family of complex two-tori, whose underlying smooth manifold is diffeomorphic to S⁶.
Alpöge–Howell–Claude: elliptic curves of rank at least 30 and 31
1 recordAlpöge and Howell report elliptic curves over the rationals with at least 30 and at least 31 independent rational points of infinite order.
An infinite finitely presented residually finite 2-group
1 recordInfinite finitely generated residually finite torsion groups were known.
Anderson delocalization at weak disorder
1 recordAbsence of eigenvalues alone does not exclude singular continuous spectrum.
Artin primitive-root infinitude for every admissible base
1 recordArtin’s conjecture predicts infinitely many primitive-root primes for every admissible fixed base, with a stronger positive-density asymptotic.
Borsuk’s covering question in dimension nine
1 recordBorsuk’s covering question asks whether a bounded set in d dimensions can be partitioned or covered by d+1 subsets of strictly smaller diameter.
Brennan’s conformal-integrability conjecture
1 recordThe target concerns the full conjectured exponent interval for arbitrary simply connected domains.
Cannon’s conjecture
1 recordA hyperbolic group with a boundary homeomorphic to the two-sphere is conjectured to act geometrically on hyperbolic three-space.
Cartan–Hadamard isoperimetry
1 recordGeodesic-ball volume comparison does not imply boundary-area comparison for arbitrary domains.
Claude reports: no infinite critical cluster on integer lattices
1 recordThe pinned Anthropic development reports θ(p_c)=0 for nearest-neighbour Bernoulli bond percolation on Zᵈ in every integer dimension d≥2.
Configuration-LP gaps in offline bin packing
1 recordOffline bin packing knows all items before assigning them to bins.
Cost-preserving unsplittable flow
1 recordCan fractional flow be rounded to one path per demand without increasing cost, while allowing the stated congestion slack?
Critical percolation on quasi-transitive graphs
2 recordsAt the critical threshold, can independently opened edges form an infinite cluster on an infinite graph with only finitely many vertex orbits?
Decidability of Medvedev logic
1 recordIs there an algorithm that decides theoremhood in Medvedev’s logic of finite problems?
Derivative-free convex oracle complexity
1 recordHow many exact function-value queries are necessary to minimize a convex Lipschitz function, at a specified dimension and accuracy?
Deterministic factorization over prime fields
1 recordA polynomial running time in p is not polynomial in the binary input length.
Embedding edit distance in ℓ₁
1 recordEdit distance counts insertions, deletions, and substitutions between strings.
Erdős problem #728: factorial divisibility
1 recordHow large can a+b−n be when a!b!
Erdős’s reciprocal-sum progression conjecture
1 recordPositive density guarantees from Szemerédi’s theorem do not automatically cover every set with divergent reciprocal sum.
Exact logarithmic-space derandomization
1 recordRestricted randomized log-space models and simulations using more than logarithmic space do not establish exact log-space derandomization.
Fermat’s Last Theorem
1 recordNo positive integers a, b, c satisfy aⁿ+bⁿ=cⁿ when the integer exponent n exceeds 2.
Formalization of 3SUM and shortest-path algorithms
1 record3SUM looks for three input numbers summing to zero; all-pairs shortest paths computes distances between every pair of graph vertices.
Fujita’s freeness conjecture
1 recordThe sharp n+1 threshold was known in several low dimensions, while general-dimensional results required larger twists.
Gauss: AI-assisted sphere-packing formalization in dimensions 8 and 24
1 recordMath, Inc.
Gauss: AI-assisted strong prime number theorem formalization
1 recordMath, Inc.
Global classical relativistic Vlasov–Maxwell solutions
1 recordThe relativistic Vlasov–Maxwell system models particles interacting with their electromagnetic field.
Goldfeld’s analytic density conjecture
1 recordSelmer-rank distributions alone do not determine analytic-rank densities.
Hilbert’s tenth problem over the rationals
1 recordUndecidability over the integers does not automatically imply undecidability over the rationals.
Infinite little finitistic dimension
1 recordThe conjecture bounds finite projective dimensions over each fixed finite-dimensional algebra.
Integer-multiplication complexity
1 recordMultiplying two n-bit integers is a basic exact computation.
Irrationality of Catalan’s constant
1 recordCatalan’s constant is the alternating sum 1 − 1/9 + 1/25 − 1/49 + … .
Jacobsthal’s function
1 recordJacobsthal’s function measures how long an interval can fail to contain an integer coprime to a number with a given count of distinct prime factors.
Kadison’s similarity conjecture
1 recordKnown results for nuclear algebras and other restricted classes do not establish the unrestricted bounded-representation target.
Kaplansky zero-divisor counterexamples
1 recordTorsion gives elementary zero divisors.
Koebe’s circle-domain conjecture
1 recordCan every connected planar domain be conformally represented by a domain whose complementary components are round disks or individual points?
Low-temperature SK free-energy fluctuations
1 recordA limiting thermodynamic free-energy density does not describe sample-to-sample fluctuations of total free energy.
Maximum matching in general graphs
1 recordA matching chooses edges without shared endpoints.
Milne’s rationality conjecture
1 recordCompatible cohomological realizations alone do not establish rational intersection numbers with new divisor classes arising after reduction.
Modularity over imaginary quadratic fields
1 recordModularity over the rationals and known special classes over other fields do not supply the unrestricted imaginary-quadratic result.
Mutually unbiased bases in dimension six
1 recordThree bases can be constructed in dimension six.
Nagata’s plane-curve conjecture
1 recordSquare values of the point count were covered by Nagata’s original cases.
Nonamenability of Thompson’s group F
1 recordThe problem asks whether F admits an invariant mean, equivalently the relevant Følner behavior.
Off-diagonal Ramsey growth
1 recordOff-diagonal Ramsey numbers R(s,t) force a clique of size s in one edge color or size t in the other.
OpenAI reports: Mahler’s conjecture for general convex bodies
1 recordFor every convex body K in R^n, n≥1, the manuscript reports volume product at its Santaló point at least (n+1)^(n+1)/(n!)², with equality exactly for simplices.
OpenAI reports: Navier–Stokes blowup with smooth forcing
1 recordFor every positive viscosity, the manuscript reports a three-dimensional incompressible Navier–Stokes flow starting from rest with smooth compactly supported forcing, bounded kinetic energy, and unbounded velocity in finite time.
OpenAI reports: a counterexample to Hadwiger’s conjecture
1 recordThe manuscript reports arbitrarily large finite simple graphs with independence number at most two and chromatic number greater than their largest clique-minor order.
OpenAI reports: a power saving for exact discrete Fourier transforms
1 recordThe manuscript reports one deterministic algorithm computing the exact complex discrete Fourier transform at every length n in O(n(log n)^(1−10^−13)) operations, including scalar preparation, scheduling and indexing, in its specified arithmetic model.
OpenAI reports: smooth unforced Euler blowup on R³
1 recordThe manuscript reports smooth compactly supported divergence-free initial data on R³ whose unforced three-dimensional incompressible Euler solution has finite maximal lifespan, with unbounded velocity gradient and divergent time integral of maximum vorticity.
Optimal-order randomized k-server
1 recordThe online problem compares movement cost with an offline optimum that sees the full sequence.
Ordinary two-point Chowla correlations
1 recordLogarithmically weighted averages and ordinary averages have different targets.
Ostmann’s inverse Goldbach conjecture
1 recordA decomposition into two nontrivial summand sets would describe the eventual prime set additively.
P versus NP
0 recordsIf a proposed solution can be checked efficiently, can a solution always be found efficiently?
Packing unit regular hexagons
1 recordPlace non-overlapping unit regular hexagons inside a larger regular hexagon.
Packing unit squares in a circle
1 recordFor a fixed number of unit squares that may rotate freely, what is the smallest enclosing circle, and which configurations attain it?
Parisi–Zamponi–Claude: an analytic proof of the jamming exponent identity
1 recordParisi and Zamponi give an analytic proof of a+b=1 from the scaling full replica-symmetry-breaking equations for infinite-dimensional hard-sphere jamming.
Planar graph metrics in L₁
1 recordA metric embedding represents distances in another space.
Resolution-over-parities proof complexity
1 recordHow long must a refutation be in a system whose clauses contain affine equations over the two-element field?
Rigidity of the Turing degrees
1 recordFixing a cone of degrees or preserving definable operations does not by itself imply that every degree is fixed.
Serre’s positivity conjecture
1 recordIntersection multiplicity is an alternating sum of lengths of Tor groups over a regular local ring.
Seymour’s second-neighborhood conjecture
1 recordThe conjecture was known for important special graph classes.
Shelah’s eventual categoricity conjecture
1 recordCategoricity means all models of a given size are isomorphic.
Short Egyptian-fraction expansions
1 recordAn Egyptian fraction expresses a rational number as a sum of distinct reciprocals of positive integers.
Shortest common superstring
1 recordA common superstring contains every input string as a substring.
Single-lattice convex-body coverings
1 recordCovering density measures overlap when translates of a shape cover space.
Smooth isometric immersion of surfaces into four-space
1 recordCan every compact smooth Riemannian surface be immersed in Euclidean four-space while preserving tangent lengths and angles?
Stochastic mean-payoff games
1 recordIn a mean-payoff game, players choose moves in a graph to optimize a long-run average reward.
Strict hot spots on simply connected planar domains
1 recordThe strict version concerns every eigenfunction, rather than existence of one favorable eigenfunction or results confined to convex or symmetric domains.
The BSD formula under low Selmer corank
1 recordEquality of analytic and algebraic ranks and finiteness of the Tate–Shafarevich group do not determine the exact leading coefficient.
The Deligne–Drinfeld conjecture
1 recordConstructing individual odd-weight elements does not establish that they freely generate the entire Lie algebra.
The Erdős unit-distance conjecture
1 recordHow many pairs among n planar points can be exactly one unit apart?
The Falconer distance conjecture
1 recordA distance set with full one-dimensional Hausdorff dimension need not have positive length.
The Gaussian moat problem
1 recordGaussian primes are primes in the integer lattice of complex numbers.
The Hilbert–Smith conjecture
1 recordMust a locally compact group acting faithfully and continuously on a finite-dimensional topological manifold be a Lie group?
The Jacobian conjecture
1 recordDoes a polynomial self-map with constant nonzero Jacobian necessarily have a polynomial inverse?
The Partition Principle without Choice
1 recordChoice implies the Partition Principle, but an injection reversing a surjection need not select a member of each fiber.
The Ramsey number R(5,5)
0 recordsFind the smallest number of vertices such that every red/blue coloring of a complete graph’s edges contains a monochromatic five-vertex clique.
The Riemann hypothesis
2 recordsThe Riemann hypothesis predicts that every nontrivial zero of the Riemann zeta function lies on the line with real part one half.
The Unique Games Conjecture
1 recordThe conjecture requires simultaneously near-perfect completeness and arbitrarily small soundness with an alphabet fixed for those errors.
The all-dimensional Gaussian propeller bound
1 recordThree-cell and three-dimensional cases were known.
The chromatic number of the plane
1 recordColor every point of the Euclidean plane so that points exactly one unit apart have different colors.
The complete Crouzeix inequality
1 recordThe numerical range controls polynomial evaluation at a matrix.
The core in approval-based committee elections
1 recordCan a selected committee resist deviations by every sufficiently large group of voters under the approval-based core definition?
The deterministic k-server conjecture
1 recordMove k servers through a metric space to serve online requests while competing with an offline optimum.
The enclosing-area spacetime Penrose inequality
1 recordThe inequality compares invariant ADM mass with the minimum area enclosing a trapped boundary in asymptotically flat initial data.
The even-periodic spin-one Haldane gap
1 recordEvidence for a gap in finite samples or a different spin-chain Hamiltonian does not establish a length-uniform gap for the pure bilinear spin-one model.
The four-dimensional Singer conjecture
1 recordPoincaré duality reduces the four-dimensional Singer target to a vanishing assertion but does not establish that vanishing.
The free group factor isomorphism problem
1 recordDoes the tracial von Neumann algebra associated with a free group remember its number of generators?
The irrationality exponent of π
1 recordThe irrationality exponent measures how closely infinitely many rational numbers can approximate a real number.
The kissing number in dimension 11
2 recordsHow many unit spheres in eleven-dimensional Euclidean space can touch one central unit sphere without overlapping each other?
The matrix-multiplication exponent
2 recordsThe exponent describes arithmetic-operation growth for large square matrices.
The pinwheel kernel conjecture
1 recordCan every feasible periodic scheduling instance be reduced to one with a universal cap on its longest period?
The quotient property of abelian envelopes
1 recordDoes every abelian envelope of a rigid monoidal category possess the quotient property?
The rational Hodge conjecture for CM abelian varieties
1 recordHodge classes are cohomological objects; the problem asks whether rational ones come from algebraic cycles on abelian varieties with complex multiplication.
The symmetric Mahler conjecture
1 recordSpecial dimensions and classes of convex bodies do not establish the all-dimensional symmetric volume-product minimum.
The three-dimensional Kakeya maximal conjecture
1 recordFull dimension of Kakeya sets and the quantitative maximal-operator estimate are different assertions.
Uniform bounds for planar polynomial limit cycles
1 recordA limit cycle is an isolated periodic orbit.
Weak inhomogeneous Duffin–Schaeffer approximation
1 recordThe shifted approximation problem fixes a real shift and asks for an almost-everywhere divergence criterion.
Worst-case Subset Sum algorithms
1 recordSubset Sum asks whether some subset of given integers has a specified sum.
Yau’s noncompact uniformization conjecture
1 recordCompact positive-curvature classification and noncompact cases with added growth or curvature assumptions do not establish the unrestricted stated pointwise-positive target.
Zero-free regions for zeta and L-functions
1 recordA zero-free region excludes zeros from a specified part of the complex plane.
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