Great problems
Ancient
- 450 BCE
- 450 BCE
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Early modern
- 1611
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18th century
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19th century
- 1844
- 1852Four colour theoremCombinatorics and discrete mathematics. Every planar map (or loopless planar graph ) can be coloured with at most colours () — posed by Francis Guthrie in 1852, proved with computer assistance by Kenneth Appel and Wolfgang Haken in 1976, simplified in 1997, and formally verified in Coq by Georges Gonthier in 2005.KAKenneth Appel, 1976
20th century
- 1900
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- 1959
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- 1985
- 1987Fundamental lemma of the Langlands programmeArithmetic and number theory, Algebra. The fundamental lemma is a combinatorial identity between orbital integrals on a reductive group and its endoscopic groups, conjectured by Robert Langlands and Diana Shelstad. It was the last missing ingredient needed to stabilize the Arthur–Selberg trace formula, the main tool linking automorphic representations across different groups. Ngô Bảo Châu proved it in general by reinterpreting the identity geometrically, as a statement about the cohomology of the Hitchin fibration.NCNgô Bảo Châu, 2008
- 1992
- 1995