Great problems

19th century

1. 1859
   Riemann hypothesisArithmetic and number theory, Analysis. The zeta function encodes the distribution of prime numbers: its zeros control the error term in the prime counting function $\pi(x)$. Riemann observed in 1859 that all the nontrivial zeros he could compute lay on the critical line $\mathrm{Re}(s)=\tfrac12$ and conjectured this always holds. A proof would give the strongest possible form of the prime number theorem; billions of zeros have been checked numerically and all lie on the line, but no proof or disproof is known. It is one of the seven Clay Millennium Prize Problems.MillenniumHilbert #8Open

20th century

1. 1904
   Poincaré conjectureTopology, Geometry. Every simply connected closed $3$-manifold is homeomorphic to the $3$-sphere $S^3$ — posed by Henri Poincaré in 1904 and proved by Grigori Perelman in 2002–2003 via Ricci flow with surgery, the only solved Clay Millennium Prize Problem.MillenniumGPGrigori Perelman, 2003
2. 1950
   Hodge conjectureGeometry, Algebra. Hodge classes are the cohomology classes that the complex structure of a variety picks out as 'shape-like'; the conjecture asks whether every one of them actually comes from a genuine algebraic subvariety. Known in codimension 1 since 1924 (Lefschetz) and for abelian varieties up to dimension 5, it is one of the seven Clay Millennium Prize Problems and remains open in general.MillenniumOpen
3. 1965
   Birch and Swinnerton-Dyer conjectureArithmetic and number theory. Predicts that the rank of an elliptic curve's rational points equals the order of vanishing of its L-function at s=1 — proven only when that order is 0 or 1.MillenniumOpen
4. 1971
   P versus NPApplied and computational mathematics, Foundations of mathematics. Asks whether every problem whose solution can be checked quickly can also be solved quickly — the central open question of computational complexity, carrying a Clay Millennium Prize.MillenniumOpen

Contemporary

1. 2000
   Navier–Stokes existence and smoothnessDifferential equations and dynamical systems. Asks whether the 3D incompressible Navier–Stokes equations always have smooth, globally defined solutions from smooth initial data, or whether they can blow up in finite time — one of the seven Clay Millennium Prize problems.MillenniumSmale #15Open
2. 2000
   Yang–Mills existence and mass gapMathematical physics, Analysis. Classical Yang–Mills fields are massless like the photon, yet in the quantum world the strong nuclear force they describe is short-range and binds gluons and quarks into massive particles. Giving a mathematically rigorous construction of four-dimensional quantum Yang–Mills theory and proving its mass gap $\Delta > 0$ is one of the seven Clay Millennium Prize Problems.MillenniumOpen