Abelian and nonabelian mathematics
type: Item
instance of: scholarly article
title: Abelian and nonabelian mathematics
zbMATH Open document ID: 0733.11043
DOI: 10.1007/BF03024075
author: I. R. Shafarevich
published in: The Mathematical Intelligencer
publication date: 1991-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R21
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R39
Mathematics Subject Classification ID: 14K30
Mathematics Subject Classification ID: 20F18
zbMATH DE Number: 4212169
zbMATH Keywords: duality theory of abelian groups
zbMATH Keywords: Galois theory of abelian extensions
zbMATH Keywords: Abelian functions
zbMATH Keywords: Class field theory
zbMATH Keywords: Langlands conjectures
zbMATH Keywords: nilpotent mathematics
zbMATH Keywords: nilextensions
zbMATH Keywords: iterated integrals
reviewed by: Helmut Koch
MaRDI profile type: Publication
Wikidata QID: Q125723614
cites work: Q5588717
cites work: Q3277020
cites work: Q5532660
cites work: Q3879354
cites work: An elementary introduction to the Langlands program
cites work: Q5619167
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description: scientific article; zbMATH DE number 4212169