Navier–Stokes existence and smoothness◌ProposedDo smooth three-dimensional incompressible Navier–Stokes flows always remain smooth, or can a finite-time singularity occur?Partial differential equations · Mathematical physicsOpenMath assessmentchecked Oct 13, 2026Partial resultsFormal sources
P versus NP○OpenCan every decision problem with polynomially bounded, polynomial-time-checkable certificates also be decided in deterministic polynomial time?Theoretical computer scienceper Clay · P versus NPchecked Oct 10, 2026
Petersen coloring conjecture✕DisprovedCan every bridgeless cubic graph be colored by the Petersen graph?Graph theoryOpenMath assessmentchecked Oct 14, 2026
Poincaré conjecture●ProvedIs every closed, simply connected three-dimensional manifold homeomorphic to the three-sphere?Topology · GeometryOpenMath assessmentchecked Oct 12, 2026Formal sources
Riemann hypothesis○OpenDo all nontrivial zeros of the Riemann zeta function have real part 1/2?Number theory · Analysisper Clay · Millennium Problemschecked Oct 10, 2026Partial results
Schiffer and Pompeiu conjectures✕DisprovedMust a bounded simply connected planar domain be a disk if it supports a nonconstant Helmholtz solution with constant boundary value and zero normal derivative?Analysis · Partial differential equations · Spectral geometryOpenMath assessmentchecked Oct 13, 2026Formal sources
Sendov’s conjecture●ProvedIf every zero of a complex polynomial lies in the closed unit disk, must each zero be within distance one of a zero of its derivative?Complex analysisOpenMath assessmentchecked Oct 13, 2026Formal sources
Strong log-concavity of the Chernoff density●ProvedDoes the Chernoff density have a uniformly positive lower bound on its negative log-curvature?Probability · StatisticsOpenMath assessmentchecked Oct 14, 2026
Three-dimensional Euler regularity◌ProposedCan smooth compactly supported initial velocity form a finite-time singularity in the three-dimensional incompressible Euler equations without external forcing?Partial differential equations · Mathematical physicsOpenMath assessmentchecked Oct 13, 2026Formal sources
Why are there infinitely many primes?●ProvedFor every natural number n, does there exist a prime p with n < p?Number theoryOpenMath assessmentchecked Oct 10, 2026Formal sourcesResearch task
Černý synchronization conjecture○OpenHow long a word is needed to bring every state of a synchronizing automaton to one state?Computer science · CombinatoricsOpenMath assessmentchecked Oct 14, 2026Partial results