Abelian and nonabelian mathematics
Many theories in the mathematics of the nineteenth and twentieth centuries are based on the duality theory of abelian groups and their groups of characters. In this connection the author speaks about abelian mathematics. He explains as examples: 1. The duality between a finite- dimensional vector space and its dual. 2. The characters of a finite abelian group. 3. Galois theory of abelian extensions. 4. Abelian functions. 5. Class field theory. - Nonabelian mathematics arises in the algebra of representations of a finite group, vector bundles and matrix divisors and the Langlands conjectures. - The main stimulus of the author for writing this survey was to draw attention to a newly emerging field between abelian and nonabelian mathematics: nilpotent mathematics. Examples are nilextensions of number fields and iterated integrals in the situation of coverings with nilpotent group of a Riemann surface [see \textit{P. Cartier}, Sémin. Bourbaki, 40ème Année, Vol. 1987/88, Exp. No.687, Astérisque 161-162, 31-52 (1988; Zbl 0688.14036)].
scientific article; zbMATH DE number 4212169
scientific article