Sophus Lie in the context of Differential equations

The Abel Prize (/ˈɑːbəl/ AH-bəl; Norwegian: Abelprisen [ˈɑ̀ːbl̩ˌpriːsn̩]) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes; as such, it is widely considered the Nobel Prize of mathematics. It comes with a medal and a monetary award of 7.5 million Norwegian kroner (NOK, about US$873,000 in 2025; increased from 6 million NOK in 2019).

The Abel Prize's history dates back to 1899, when its establishment was proposed by the Norwegian mathematician Sophus Lie when he learned that Alfred Nobel's plans for annual prizes would not include a prize in mathematics. In 1902, King Oscar II of Sweden and Norway indicated his willingness to finance the creation of a mathematics prize to complement the Nobel Prizes, but the establishment of the prize was prevented by the dissolution of the union between Norway and Sweden in 1905. It took almost a century before the prize was finally established by the Government of Norway in 2001, and it was specifically intended "to give the mathematicians their own equivalent of a Nobel Prize." The laureates are selected by the Abel Committee, the members of whom are appointed by the Norwegian Academy of Science and Letters.

In mathematics, the mathematician Sophus Lie (/liː/ LEE) initiated lines of study involving integration of differential equations, Automorphism group and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan.

The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data.