Great problems

20th century

1. 1900
   Hilbert's sixteenth problemDifferential equations and dynamical systems, Geometry. Asks for the possible arrangements of ovals of a degree-$n$ real algebraic curve and for a uniform bound $H(n)$ on the number of limit cycles of a degree-$n$ polynomial vector field in the plane. Every individual polynomial system has finitely many limit cycles (Ilyashenko 1991; Écalle 1992), but whether $H(n) < \infty$ remains open even for $n = 2$.Hilbert #16Smale #13Open
2. 1939
   Jacobian conjectureAlgebra, Analysis. Asks whether every polynomial map $k^n \to k^n$ in characteristic zero with constant nonzero Jacobian determinant has a polynomial inverse; disproved for $n \ge 3$ in 2026 while the planar case $n = 2$ remains open.Smale #16Partially solved
3. 1963
   Existence of the Lorenz attractorDifferential equations and dynamical systems. Proves that the classical Lorenz differential equations genuinely possess a robust chaotic strange attractor, resolving Smale's 14th problem via a rigorous computer-assisted proof by Warwick Tucker.Smale #14Solved

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