Millennium problems

• P vs NP

  P = NP? That is: can every decision problem whose yes-instances have polynomial-time verifiable certificates also be decided in polynomial time? Equivalent formulations run through NP-completeness (Cook–Levin) and Karp reductions.

• Hodge

  On a non-singular complex projective variety, every Hodge class (a rational (p,p)-class in Betti cohomology) is a rational linear combination of classes of algebraic cycles.

• Poincaré

  Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. Settled by Perelman (2002–03) via Ricci flow with surgery, entropy monotonicity, and no collapsing.

• Riemann

  Every non-trivial zero of the Riemann zeta function ζ(s) has real part 1/2. Equivalent forms involve the ξ-function, the explicit formula, and zero-density estimates on the critical strip.

• Yang–Mills

  For any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has a mass gap Δ > 0. Construct the theory in a mathematically complete way (Wightman or equivalent) and prove the gap.

• Navier–Stokes

  In three space dimensions and time, given a smooth divergence-free initial velocity of finite energy, do there exist a smooth velocity and pressure, globally defined, solving the incompressible Navier–Stokes equations? Prove existence and smoothness, or exhibit a blow-up.

• BSD

  For every elliptic curve E over ℚ, the rank of the Mordell–Weil group E(ℚ) equals the order of vanishing of the Hasse–Weil L-function L(E,s) at s = 1. Finiteness of Sha is part of the expected picture.