Collected mathematical papers. Volume 1 (1913-1932). Volume 2 (1932- 1955). Ed. for the Danish Mathematical Society by Vagn Lundsgaard Hansen.

*(English)*Zbl 0609.01050
Contemporary Mathematicians. Boston-Basel-Stuttgart: BirkhĂ¤user. Vol. 1: XI, 459 p. DM 248.00; Vol. 2: XI, 441 p. DM 240.00 (1986).

Jakob Nielsen (1890-1959), one of the central figures of Danish mathematics in the first half of this century, is well known throughout the international mathematical community for his pioneering work and remarkable achievements in combinatorial group theory and the topology of surfaces. The recent revival of the theory of surface transformations has led to a renewed strong interest in his original methods and results. The decision taken by the Danish Mathematical Society to make his original papers available to a wider audience must therefore be greeted with satisfaction. These two volumes contain a selection of 33 from Nielsen’s 46 published papers (two of them coauthored by S. Bundgaard and one by W. Fenchel) and also a set of notes from his inaugural lectures in Breslau in 1921. From these, 11 papers, including Nielsen’s dissertation (presented to the University of Kiel in 1912 and published in 1913), appear in their original German version; 7 papers, originally published in German and including the Breslau notes and his four long memoirs in ”Acta Mathematica” from 1927 [Acta Math. 50, 189-358 (1927; JFM 53.0545.12)], 1929 [ibid. 53, 1-76 (1929; JFM 55.0971.01)], 1932 [ibid. 58, 87-167 (1932; Zbl 0004.27501)] and 1942 [ibid. 75, 23-115 (1942; Zbl 0027.26601)] which present his pioneering work on surface transformations, appear here in John Stillwell’s English translation; other 10 papers, originally published in Danish and including the famous paper on free groups from 1921, also appear in English translation (one by Anne W. Neumann, one by Hans J. Munkholm, and the others by J. Stillwell); the remaining 6 papers (one in French and 5 in English) are reproduced in their original version. Volume 1 opens with Werner Fenchel’s memorial paper which presents Nielsen’s life and mathematical achievements; it includes a complete list of Nielsen’s publications. Volume 2 concludes with the following 5 essays: Bruce Chandler and Wilhelm Magnus - ”The group theoretical papers of Jakob Nielsen” (adaptation of Chapter II. 2 of the book ”History of combinatorial group theory. A case study of the history of ideas (1982; Zbl 0498.20001); Joan Birman - ”Nielsen’s investigation of surface mapping classes”; Heiner Zieschang - ”On Nielsen’s work on finite mapping classes and related subjects”; Robert F. Brown - ”Nielsen fixed point theory”; Werner Fenchel - ”Nielsen’s contributions to the theory of discontinuous groups of isometries of the hyperbolic plane”. These essays discuss in detail Nielsen’s contributions to these subjects and their impact on modern mathematical research.

Reviewer: J.Weinstein