Back Hilberta kvina problemo Esperanto Quinto problema de Hilbert Spanish Cinquième problème de Hilbert French הבעיה החמישית של הילברט HE Quinto problema de Hilbert Portuguese Пятая проблема Гильберта Russian Hilberts femte problem Swedish П'ята проблема Гільберта Ukrainian 希尔伯特第五问题 Chinese
Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups.
The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics (for example quark theory) grew steadily in the twentieth century. In rough terms, Lie group theory is the common ground of group theory and the theory of topological manifolds. The question Hilbert asked was an acute one of making this precise: is there any difference if a restriction to smooth manifolds is imposed?
The expected answer was in the negative (the classical groups, the most central examples in Lie group theory, are smooth manifolds). This was eventually confirmed in the early 1950s. Since the precise notion of "manifold" was not available to Hilbert, there is room for some debate about the formulation of the problem in contemporary mathematical language.