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UNIVERSITY OF COPENHAGEN
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Werner Fenchel papers


General, short description of the collection

A short description:
This collection of Werner Fenchel papers consists of a large number of mathematical manuscripts, reprints of almost all of Fenchel's scientific publications, some material concerning Jakob Nielsen, some biographical material and a little correspondance. The mathematical manuscripts are a mixture of, on the one hand, manuscripts for lectures, and on the other hand, manuscripts of scientific works. Among other things there are several versions of a manuscript of a book. Unfortunately, most of the manuscripts are undated and several of them are without a title. The material is from the period 1908-86 and there is approximately the same amount of material in Danish, in German and in English. There is also a small amount of material in French or Italian.

Size:
The collection is contained in 12 archival boxes.

Handed over:
Originally a group of the staff of the Mathematics Institute got permission from Tom Fenchel (the son of Werner Fenchel) to sort out what they found worth keeping of what Werner Fenchel left, when he died in 1988. The collection became the result of that effort. The collection had been kept in the basement of the mathematical library in the E-block of the H. C. Ørsted Institute until the fall of 1996, when it was moved to the archive.

Clauses:
No.

Copyright:
The copyright of the manuscripts and other material produced by Werner Fenchel belongs to Tom Fenchel, while the copyrights of any letter belongs to the author of the letter and his/her descendants.

Inventory:
This inventory was made by Sigurd Elkjær and Kurt Ramskov 20 August 1998 and translated into English by Ramskov in October 1998. Box 13 added in 2003.

Further information on archival material etc.:
Theseus homepage of Werner Fenchel.



A short biography of Werner Fenchel

Werner Fenchel (1905-88) was born in Berlin. He studied physics and mathematics at the University of Berlin, and he decided to concentrate on mathematics. In 1928 he submitted his doctoral dissertation about closed space curves. His first academic position was as an assistant to professor E. Landau at the University of Göttingen. During the years 1930-32 he went abroad several times on a Rockefeller fellowship. He stayed in Italy and he also went to Denmark for a period to collaborate with Bonnesen. When the Nazis took over power in 1933 many Jews in the state administration, including the universities, soon lost their jobs. Because Fenchel was of Jewish descent he was also affected, and he left for Denmark. In December 1933 he married Käte Sperling, who also was a mathematical refugee. They managed to get things going by having temporary jobs. They both felt at home among the Danish mathematicians, and Werner Fenchel collaborated with several of them. As most other Jews in Denmark they succeeded in escaping to Sweden during 1943-45. During that period they were both teaching at the Danish School in Lund. In 1947 Fenchel got a position in mathematics at the Technical University. He became professor in rational mechanics the same place in 1951 and in 1956 he succeeded Nørlund as professor in mathematics at the University of Copenhagen. Fenchel's scientific work was mainly in two subjects. In the early period he worked on the theory of convexity, while in his later years, after World War II, he worked on a theory of discontinuous groups of motion in the non-Euclidean plane. He was involved in the publication of the collected works of Jakob Nielsen.



A survey of the content of the collection

THe Werner Fenchel papers have been divided into subjects and the material belonging to a subject has then, as far as possible, been sorted chronologically. Unfortunately, a lot of the material is undated and for a major part of the manuscripts it has not been possible to decide whether it was the manuscript for a lecture or a draft for a scientific work. Hence the division into "scientific manuscripts" and "manuscripts for lectures" is not very sharp, i.e. one may find manuscripts for lectures in the subject "scientific manuscripts".

Subject Boxes
Biographical material and correspondence 1
Scientific manuscripts 1-5
Manuscripts for lectures 6
Notes 7-8
Publications 9-10
Material concerning Jakob Nielsen 10
Miscellaneous 11-13



List of the content of each of the boxes

BIOGRAPHICAL MATERIAL AND CORRESPONDENCE

Box 1
Folder: Memories of the time of study
Folder: Press clippings
Folder: His 70th anniversary
Folder: Correspondence from Jakob Nielsen 1939-59
Folder: Miscellaneous correspondence


SCIENTIFIC MANUSCRIPTS

Box 1 (continued)
Two notebooks about determinants

Box 2
Folder: On convex optimization problems [>1963]
Folder: Konvexität und Dualität [>1976]
Folder: Manuscripts on convex sets and functions
Folder: Manuscript on homology and cohomology
Folder: Riemannsk geometri [Riemannian geometry]
Envelope: Überlagungsfläche und Fundamentalgruppe

Box 3
Folder: Manuscript on discontinuous groups of motion
Sections of a book manuscript on "Discontinuous groups and motions", 1

Box 4
Sections of a book manuscript on "Discontinuous groups and motions", 2

Box 5
Sections of a book manuscript on "Discontinuous groups and motions", 3
Envelope: W. Fenchel: Introduzione alla teoria dei gruppidiscontinui di trasformazioni
W. Fenchel: Introduzione alla teoria dei gruppi discontinui di trasformazioni 1956
Folder: Miscellaneous manuscripts


MANUSCRIPTS FOR LECTURES

Box 6
Plastic folder: W. Fenchel: Krystallografiske grupper
Plastic folder: Diskontinuerte transformationsgrupper
Plastic folder: W. Fenchel: On trigonometry in hyperbolic 3-space
Plastic folder: Über diskontinuierliche Isometriegruppen der nicht-euklidischen Ebene und ein Problem der Flächentopologie
Plastic folder: Transparencies in German (8 pieces).
Plastic folder: Transparencies in German for non-Euclidean Geometry (12 pieces).
Folder: Lecture on topology of surfaces
Folder: Manuscripts for lectures 1940-1963
Folder: Notes and manuscripts for lectures on topology 1954-56
Folder: Manuscript for lecture: Ikke-euklidisk geometri 1964.


NOTES

Box 7
W. Fenchel: Konvekse Mængder og Funktioner 1937/38
W. Fenchel: Convex Bodies 1951
W. Fenchel: Convex cones, sets, and functions 1951 (3 copies)
W. Fenchel: Om differential- og integralregningens opbygning 1956

Box 8
W. Fenchel: Om aritmetikkens grundlag 1957 (3 copies)
W. Fenchel: Conjugate convex functions 1964 (5 copies)
Seminar over konvekse mængder og funktioner 1965
Folder: Notes on topology and groups
Minkowskis polaritet


PUBLICATIONS

Box 9
Plastic folder: Two bibliographies
Folder: Reprint of articles 1929-47
Folder: Reprint of articles 1948-83 etc.
Folder: Encyclopedia articles
T. Bonnesen und W. Fenchel: Theorie der Konvexen Körper 1934 (2 copies)

Box 10
Werner Fenchel (ed.): Proceedings of the Colloquium on Convexity, Copenhagen 1965. Københavns Universitets Matematiske Institut 1967.


MATERIAL CONCERNING JAKOB NIELSEN

Box 10 (continued)
Folder: Jakob Nielsen manuscripts 1921
Folder: Jakob Nielsen's collected works
Manuscript: Vagn Lundsgaard Hansen: En dansk matematikers eventyr
Obituary of Jakob Nielsen
Folder: Jakob Nielsen reprints


MISCELLANEOUS

Box 11
Plastic folder: G. Fubini: Introduzione alla teoria dei gruppé discontinué e delle funzioni automorfe 1908
Georg Hajós: Über einfache und mehrfache Bedeckung des n- dimensionalen Raumes mit einem Würfelgitter. Photocopy of pp. 427-67 of a journal from 1941.
Ladislaus Rédei: Neuer Beweis des Hajósschen Satzes über die endlichen Abelschen Gruppen. Photocopy of pp. 27-39 of a journal.
L. Rédei: Die neue Theorie der Endlichen Abelschen Gruppen und Verallgemeinerung des Hauptsatzes von Hajós. Photocopy of pp. 329-373 of a journal.
John Stillwell: Poincaré, Dehn, and the Origins of Nielsen's Theory of Surfaces. Typewritten manuscript, 38 pp.
John C. Stillwell: Translation of Dehn's 1922 Breslau Lecture. Paper No. 1, October 1981. Department of Mathematics, Monash University, Australia.
Vagn Lundsgaard Hansen: En geometer ser på naturen. Photocopy of typewritten manuscript, 24 pp.
Allan Sandqvist: En forbindelse mellem sædvanlige ikke-lineære differentialligninger af 1. orden og variationsregning. Mat-report No. 1986-05, Matematisk Institut, Danmarks Tekniske Højskole, Februar 1986.
Norbert Schappacher: Das Mathematische Institut der Universität Göttingen 1929-1950. Typewritten, in German, 43 pp.
Menniche m.fl.: Typewritten notes on among other things Folder: Miscellaneous

Box 12
A small black box with card files containing name and address of colleagues etc. On the back of each card is a list of reprints sent to the person. There is also some names in the box, which are not on a card.

Box 13
Material from the publication of Werner Fenchel and Jakob Nielsen: Discontinuous Groups of Isometries in the Hyperbolic Plane, edited by Asmus L. Schmidt, De Gruyter Studies in Mathematics Vol. 29, De Gruyter, Berlin 2003.



Details about the content of some of the folders

Box 1

Folder: Memories of the time of study
Typewritten manuscript in Danish for a talk (probably a talk in the Danish Mathematical Society), 18 pp. Without title or date. The content looks partly like the content of Fenchel's article: "Erinnerungen aus der Studienzeit", Jahrbuch überblicke Mathematik, Bibliographisches Institut, Mannheim 1980, pp. 155-166.

Folder: Press clippings
Press clippings about Werner Fenchel and other mathematicians and scientists. The articles about Fenchel include the following:
  • Den politiske Flygtning paa Universitetet, Berlingske Aftenavis, Torsdag 3. Marts 1955.
  • Matematik-Professorat besættes uden Opslag. Berlingske Tidende, Mandag den 28. februar 1955.
  • Blev kaldet til Universitetet. Konsistorium indstiller Professor Werner Fenchel som Professor N. E. Nørlunds Efterfølger. [1955]

Folder: His 70th anniversary
Material from Werner Fenchel's 70th anniversary in 1975: received letters of congratulation, reprint of an article by U. Dieter dedicated to Fenchel on his 70th anniversary, songs and an advertisment for Sølyst in Klampenborg.

Folder: Correspondence from Jakob Nielsen 1939-59
Below is a list of the letters, which are all from Jakob Nielsen to W. Fenchel. If nothing else is said they are handwritten, in Danish and only 1 page long.
  • 10 November 1939 (in German, 4 pages)
  • 21 December 1950
  • 1 June 1953 (Enclosed are handwritten drafts in German for paragraph 21.4-6, 21.1 and 22 and drafts in English for paragraph 21 and 22 of Discontinuous groups and motions with Nielsen's comments and corrections. However, all drafts do not necessarily belong to this letter)
  • 8 May 1954 (Enclosed is the manuscript for the lecture Om Flytningsgrupper i den hyperbolske Plan og periodiske Fladeafbildninger [On groups on motion in the hyperbolic plane and periodic maps on surfaces]. It is handwritten by Fenchel in Danish and it is 13 pages long. Also the manuscript Über Abbildungsklassen endlichen Ordnung, handwritten by Fenchel, in German and 4 pages long, is enclosed), 12 May 1954 (2 pages, put together with the previous letter)
  • 8 June 1954
  • 18 August 1954
  • 30 August 1954 (The letter is typewritten and enclosed is a handwritten draft in English for paragraph 19 and a typewritten manuscript in English for paragraph 19-21 of Discontinuous groups and motions with Nielsen's comments and corrections)
  • 29 January 1957
  • 12 March 1957
  • 22 February 1958 (Enclosed is the table of content of Discontinuous groups and motions)
  • 29 June 1958
  • 31 May 1959 (The letter is addressed to Käte Fenchel, 2 pages)

Folder: Miscellaneous correspondence
A letter from W. Fenchel to Børge Jessen, 11 December 1931. It is typewritten, 3 pages long and with a color illustration included.
Letters in English from Donald W. Blackett, Harold Kuhn and A. W. Tucker from the period 1951-53. The letters are mainly about proof-reading of Fenchel's lecture notes. Furthermore, there is the carbon copy of a letter from Fenchel to Kuhn. There are 2 letters from Blackett, 2 from Kuhn and 11 from Tucker.
Carbon copies of 2 letters in English from W. Fenchel to Gerstenhaber from 1954 concerning Fenchel's review of a work by Gerstenhaber (the review is also included), and a letter in English from Gerstenhaber to Wehausen from 1954 on the same subject.
A postcard in German from Otto Haupt to W. and K. Fenchel from 1967.
A letter in German from Annemarie Ulm to W. and K. Fenchel from 1983.
A letter from Vagn Lundsgaard Hansen to W. Fenchel from 1982.

Two notebooks on determinants
Two bounded black notebooks in A5 format with handwritten notes in German on determinants. There are in total 142 pages. It is undated and without an author.

Box 2

Folder: On convex optimization problems [>1963]
Handwritten, undated manuscript in English in Fenchel's handwriting. Perhaps for a lecture. It is written after 1963.

Folder: Konvexität und Dualität [>1976]
W. Fenchel: Konvexität und Dualität. A typewritten, undated manuscript in German 24 pages long. Because of the references it must be written after 1976.

Folder: Manuscripts on convex sets and functions
Bidrag til de konvekse funktioners teori [Contribution to the theory of convex functions]. A typewritten manuscript in Danish without date or author. 7 pp.
Two handwritten manuscripts in German in Fenchel's handwriting without title, date and author. Each is 5 pages long.
Duality in the theory of convex functions. Handwritten manuscript in English in Fenchel's handwriting without date and author. 20 pp.
Two handwritten manuscripts in English in Fenchel's handwriting without title, date and author. They are respectively 5 and 3 pages long.
A simple solution of the isoperimetric problem. Handwritten manuscript in English in Fenchel's handwriting without author and date.
Det isoperimetriske problem [The isoperimetic problem]. Handwritten manuscript in Danish in Fenchel's handwriting without author, or date. 9 pp.
Handwritten manuscript in Danish in Fenchel's handwriting on convex sets without title, author and date. 6 pp.
Kap. VIII. Konvekse Funktioner [Convex functions]. Handwritten draft for a manuscriptin Danish in Fenchel's handwriting. There are 39 numbered pages followed by 19 numbers pages entitled: Kap. N+1: Støttefunktion [Support function] and Kap. N+2: Polaritet[Polarity], a chapter about volume determination on 13 numbered pages, a chapter entitled Tschebyscheff-Approximationen on 7 numbered pages and, finally, a chapter on n dimensional spaces on 19 pages.

Folder: Manuscript on homology and cohomology
There are handwritten pages in Danish in plastic folders divided into the following sections: Paragraf 1. Sammenhængende topologiske rum [Connected topological spaces].
Paragraf 2. Fundamentalgruppe [The fundamental group].
Paragraf 2. Overlejringsrum.
Eksempler [Examples].
H-rum [H spaces].
Numerabel basis [Countable basis].
Mangfoldigheder [Manifolds].
Deling af enheden [Division of the unit].
Vektorfelter [Vector fields].
Riemann mangfoldigheder [Riemann manifolds].
Orientering [Orientation].
Differentialformer [Differential forms].
Singulær homologi [Singular homology].
Stokes' sætning [Stokes' theorem].
Differentiabel singulær homologi [Differentiable singular homology].
G. de Rham cohomologi [G. de Rham cohomology].
Riemann flader [Riemann surfaces].
Hilbertrummet af differentialformer [The Hilbert space of differential forms].
Uniformisering [Uniformization].
Litteratur [Literature].

Folder: Riemannsk geometri [Riemannian geometry]
Handwritten, undated manuscript in Danish (and afterwards in English) in Fenchel's handwriting.

Envelope: Überlagungsfläche und Fundamentalgruppe
Typewritten, undated manuscript in German with handwritten corrections; divided into 14 sections and 55 pages long.

Box 3

Folder: Manuscript on discontinuous groups of motions
Typewritten manuscript (with formulas added in handwriting) in German consisting of two chapters: Kapitel I: Kreisverwandtschaften un nichteuklidische Geometrie and Kapitel II: Diskontinuierliche Bewegungsgruppen. There are 126 numbered pages. Without author and date. It is probably an early version of the first two chapters of the work Discontinuous Groups of Motions....

Sections of a book manuscript on "Discontinuous groups and motions", 1
Typewritten manuscript in English with handwritten corrections. There are manuscripts for the following sections: the index, 26, the table of contents, 1-26, 18. The sections have be collected into the following four chapters:
  1. Möbius Transformations and Non-Euclidean Geometry.
  2. Discontinuous Groups of Motions and Reversions.
  3. Surfaces connected with discontinuous Groups.
  4. Decomposition of Groups.


Box 4

Sections of a book manuscript on "Discontinuous groups and motions", 2
Typewritten manuscript in English with handwritten corrections. There are manuscripts for the following sections: 27, 17, 19, 24, 17, 19, 21, 23, 22, pp. 120-25, the table of contents, 1-27. The sections have be collected into the following four chapters:
  1. Möbius Transformations and Non-Euclidean Geometry.
  2. Discontinuous Groups of Motions and Reversions.
  3. Surfaces connected with discontinuous Groups.
  4. Decomposition of Groups.


Box 5

Sections of a book manuscript on "Discontinuous groups and motions", 3
Typewritten manuscript in English with handwritten corrections. There are manuscripts for the following sections (some only partial): 21, 19, 18, the index,17, 26, 25, 24, 23, 22, 21, 20, 23, 22, 21, and pp. 148-215 in a pagenumbered version. The sections have be collected into the following four chapters:
  1. Möbius Transformations and Non-Euclidean Geometry.
  2. Discontinuous Groups of Motions and Reversions.
  3. Surfaces connected with discontinuous Groups.
  4. Decomposition of Groups.

Envelope: W. Fenchel: Introduzione alla teoria dei gruppi discontinui di trasformazioni
Typewritten manuscript in Italian with corrections 51 pages long.

W. Fenchel: Introduzione alla teoria dei gruppi discontinui di trasformazioni 1956
Roma - Istituto Matematico dell'Università, 1956. Typewritten copy in Italian 50 pages long.

Folder: Miscellaneous manuscripts
  • Werner Fenchel: On total curvatures of Riemannian manifolds. I. Typewritten manuscript in English with formulas included in handwriting. 11 pp.
  • Bemærkninger til M. Pihl, Finslers rum og den klass. mek.[Remarks to M. Pihl, Finsler's space and the classical mechanics]. Handwritten remarks in Danish in Fenchel's handwriting. Undated. 4 pp.
  • Handwritten manuscript for a lecture in French in Fenchel's handwriting on Fuchsian groups. Without date and title. 24 pp.
  • Handwritten manuscript in German in Fenchel's handwriting on Fuchsian groups without date or title. 8 pp.
  • Handwritten manuscript in English in Fenchel's handwriting on discontinuous groups of motion without date, author or title. 29 pp.
  • Hyperbolsk trigonometri [Hyperbolic trigonometry]. A copy of handwritten manuscript in Danish without date and author. 4 pp.
  • Chapter I: Möbius Transformations and Non-Euclidean Geometry. Handwritten manuscript in English without author or date. A version 6 pages long (and a copy of it), a version 13 pages long and a version 21 pages long.
  • Introduction to the theory of discontinuous groups. Handwritten manuscript in English in Fenchel's handwriting without date or author.
  • Handwritten manuscript in English in Fenchel's handwriting on discontinuous groups of motion without date, author or title.
  • Miscellaneous manuscripts and notes without title, date or author.


Box 6

Plastic folder: W. Fenchel: Krystallografiske grupper [Crystallographic groups]
Handwritten, undated manuscript in Danish 28 pages long. Probably for a lecture.

Plastic folder: Diskontinuerte transformationsgrupper [Discontinuous groups of transformation]
Handwritten, undated manuscript for a talk in Danish in Fenchel's handwriting 16 pages long and with 9 transparencies.

Plastic folder: W. Fenchel: On trigonometry in hyperbolic 3-space
Typewritten, undated manuscript for a talk in English 11 pages long and with 9 transparencies.

Plastic folder: Über diskontinuierliche Isometriegruppen der nicht-euklidischen Ebene und ein Problem der Flächentopologie
Handwritten, undated manuscript for a talk in German on approximately 20 pages.

Folder: Lecture on topology of surfaces
Typewritten manuscript in German entitled: Erster Abschnitt: F-Gruppen. Undated.

Folder: Manuscripts for lectures 1940-1963
There are manuscripts and often also the announcement of the following lectures (talks) presented by Werner Fenchel:
  • Om Hyperblen og Relativitetsteorien [On the hyperbola and the theory of relativity]. Organized by "Parentesen" 4 October 1940. The announcement and a handwritten manuscript in Danish 19 pages long.
  • Om fullkomliga optiska instrument [On perfect optic instruments]. Organized by Lunds MatematiskaSällskap, 26 November 1943. The announcement and a handwritten manuscript in Danish 10 pages long plus an additional 5 pages draft.
  • The announcements of meetings in Copenhagen 15-17 October 1951 respectively 17-21 October 1955 in Foreningen af matematiklærere ved gymnasieskoler og seminarier [The Association of mathematics teachers] (where Fenchel gave talks).
  • Lukkede kurver [Closed curves]. Organized by Svenska Matematikersamfundet [The Swedish Mathematical Society], 22 March 1952. The announcement and a handwritten manuscript in Danish.
  • Om generalisering av den sfäriska trigonometrien [On generalisations of the spherical trigonometry]. Organized by Lunds Matematiska Sällskap, 3 March 1952. The announcement of the talk.
  • Duale tal [Dual numbers]. Organized by "Parentesen" 19 February 1954. Handwritten manuscript in Danish 4 pages long.
  • Om konvekse mængder, overdækninger o.l. [On convex sets, coverings etc]. Organized by "Parentesen"22 February 1957. The announcement and a handwritten manuscript in Danish 13 pages long.
  • Manuscript for a talk on linear programming in Fenchel's handwriting. Undated and in Danish. 16 pp. There are 6 additional pages on applications.
  • Programming problem. Handwritten, undated manuscript in German in Fenchel's handwriting 15 pages long.
  • Ein Dualitetsprinzip für konvexe Funktionen. Handwritten, undated manuscript in German in Fenchel's handwriting 6 pages long.
  • Om orientering af linie, plan og rum [On orientation of line, plane and space], 31 October 1963. Handwritten manuscript in Danish in Fenchel's handwriting 12 pages long.
  • Relationen mellem antallet af bjerge, dale, pas og søer [The relation between the number of mountains, valleys, passes and lakes]. Organized by Lunds Matematiska Sällskap, 8 November 1963. The announcement and a handwritten manuscript in Danish 5 pages long.
  • Über die Grundlagen der affinen Geometrie. Handwritten, undated manuscript in German in Fenchel's handwriting together with a manuscript in Danish.
  • Lidt om Cirklen [A bit about the circle]. Handwritten, undated manuscript in Danish in Fenchel's handwriting 5 pages long.
  • Integralgeometri [Integral geometry]. Handwritten, undated manuscript in Danish in Fenchel's handwriting 6 pages long plus an additional draft 10 pages long.
  • Om Gitterpunkter og Kædebrøker [On lattice points and continued fractions]. Handwritten, undated manuscript in Danish in Fenchel's handwriting 9 pages long.
  • Handwritten, undated manuscript in English without title in Fenchel's handwriting 5 pages long.
  • Handwritten, undated manuscript in Danish on groups of transformation without title in Fenchel's handwriting, 5 pages long.
  • Handwritten, undated manuscript in English without title on the Gauss-Bonnet theorem in Fenchel's handwriting 3 pages long.
  • Miscellaneous fragments of manuscripts.

Folder: Notes and manuscripts for lectures on topology 1954-56
Elementær topologi [Elementary topology], Spring 1956. Handwritten manuscript for lecture in Fenchel's handwriting. 22 pages plus 2 pages of bibliography for "Elementary topology" + 5 pages entitled Klassifikation af flader [Classification of surfaces].
Lectures by A. W. Tucker: Topology. Summer Conference for High School Teachers of Mathematics, University of Washington, Seattle, July 26 - August 20, 1954. Fenchel's copy of typewritten stencils (38 pages long with 19 pages of exercises) divided into the following sections:
Lecture 1: Elementary Ideas (Notes by Carl B. Allendoerfer, pp. 1-3)
Lecture 2: The Topological Anatomy of Surfaces (Notes by Richard Long, pp. 4-12)
Lecture 3: Continuous Mappings and Vector Fields (Notes by Robert Ives, pp. 13-18)
Lecture 4: Continuous Mappings and Vector Fields (Notes by Richard Long, pp. 19-28)
Lecture 5: Manifolds, Products, and Fiberings (Notes by Robert Ives, pp. 29-38)

Dr. Hopf: Vorlesungen über Topologie. 168 typewritten pages of stencils.

Folder: Manuscript for lecture: Ikke-euklidisk geometri 1964
Handwritten manuscript in Danish for a lecture on non-Euclidean geometry from the spring term of 1964.

Box 7

W. Fenchel: Konvekse Mængder og Funktioner 1937/38
Handwritten notes in Danish entitled Convex Sets and Functions. 112 pages.

W. Fenchel: Convex Bodies 1951
Offered at Stanford University, Summer 1951. Notes by G. Hodrick. 116 pages typewritten notes.

W. Fenchel: Convex cones, sets, and functions 1951 (3 copies)
W. Fenchel: Convex cones, sets, and functions. From notes by D. W. Blackett of lectures at Princeton University, Spring Term 1951. September 1953. A list of errors is included.
W. Fenchel: Convex cones, sets, and functions. From notes by D. W. Blackett of lectures at Princeton University, Spring Term 1951. September 1953. With a list of errors included in the end. (2 copies of which one has an additional list of errors).

W. Fenchel: Om differential- og integralregningens opbygning 1956
W. Fenchel: Om differential- og integralregningens opbygning [On the structure of the differential and integral calculus]. Talk at meeting for teachers at teacher colleges 3-14 January 1956. 85 pages of typewritten notes.

Box 8

W. Fenchel: Om aritmetikkens grundlag 1957 (3 copies)
Notes in Danish entitled On the foundation of arithmetic for a talk presented at a meeting of the Association of mathematics teachers (Matematiklærerforeningen) organized for Copenhagen and suburbs in the fall of 1957. Typewritten notes. 3 copies of which two are bounded.

W. Fenchel: Conjugate convex functions 1964 (5 copies)
International Summer School on Non Linear Programming, Menton (France), 21th June - 3rd July, 1964. A note in English on 13 typewritten pages. 5 copies.

Seminar over konvekse mængder og funktioner 1965
Typewritten note in Danish in 8 pages from the fall term of 1965.

Folder: Notes on topology and groups
Handwritten manuscript in Danish on topology and groups without date, author or title. It consists of the following 9 chapters:
  • Topologiske rum [Topological spaces] (10 pages)
  • Topologiske grupper [Topological groups] (11 pages)
  • Eksempler [Examples] (19 pages)
  • Transformationsgrupper [Transformation groups] (12 pages)
  • Diskontinuerte transformationsgrupper [Discontinuous groups of transformation] (13 pages)
  • Endelige orthogonale grupper [Finite orthogonal groups] (12 pages)
  • Krystallografiske grupper [Crystallographical groups] (37 pages)
  • Diskontinuerte grupper af isometrier [Discontinuous groups of isometries] (9 pages)
  • Homogene rum for lineære og projektive grupper [Homogeneous spaces for linear and projective groups] (68 pages)

Minkowskis polaritet
Typewritten note in Danish in 5 pages. There are 3 copies.

Box 9

Plastic folder: Two bibliographies
Handwritten bibliography of W. Fenchel's publications containing 38 titles (only titles).
Typewrittne and handwritten bibliography of W. Fenchel's publications with complete reference. It contains 47 works.

Folder: Reprints of articles 1929-47
There are reprints of the following articles (if there are more than one copy, it is mentioned):
  1. Über Krümmung und Windung geschlossener Raumkurven. Mathematische Annalen 101 (1929), pp. 238-252. (Inaugural-Dissertation, Friedrich-Wilhelms-Universität, Berlin 1929).
  2. Geschlossene Raumkurven mit vorgeschriebenem Tangentenbild. Jahresbericht der Deutschen Mathematiker-Vereinigung 39 (1930), pp. 183-185.
  3. Über analytische Funktionen, die in Teilgebieten des Einheitskreises beschränkt sind. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1930), pp. 124-132.
  4. Bemerkungen über die im Einheitskreis meromorphen schlichten Funktionen. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse 23 (1931), pp. 431-436.
  5. Sulle onde di canale di tipo permanente. Rendiconti della R. Accademia nazionale dei Lincei, Classe di Scienze fisiche, matematiche e naturali (6), 13 (1931), pp. 740-743.
  6. Elementare Beweise und Anwendungen einiger Fixpunktsätze. Matematisk Tidsskrift, Afdeling B (1932), pp. 66-87.
  7. Über einen Jacobischen Satz der Kurventheorie. The Tôhoku Mathematical Journal 39 (1934), pp. 95-97. (2 copies with added comments).
  8. With B. Jessen: Über fastperiodische Bewegungen in ebenen Bereichen und auf Flächen. Det Kongelige Danske Videnskabernes Selskab, Mathematisk-Fysiske Meddelelser no. 6, 13 (1935), 28 pp.
  9. Anhang til Otto Frhr. v. Mering. Der Geltungsbereich einiger neuerer Schutzzollargumente. Archiv für matematische Wirtschafts- und Sozialforschung 1 (1935), pp. 156-158.
  10. With H. Bohr: Ein Satz über stabile Bewegungen in der Ebene. Det Kongelige Danske Videnskabernes Selskab, Mathematisk-Fysiske Meddelelser no. 1, 14 (1936), 15 pp.
  11. Über beschränkte lineare Gruppen. Matematisk Tidsskrift, Afdeling B (1936), p. 10. (2 copies of which one contains corrections).
  12. Inégalités quadratiques entre les volumes mixtes des corps convexes. Comptes rendus hebdomadaires des séances de l'Académie des Sciences (Paris) 203 (1936), pp. 647-650.
  13. Généralisation du théorème de Brunn et Minkowski concernant les corps convexes. Comptes rendus hebdomadaires des séances de l'Académie des Sciences (Paris) 203 (1936), pp. 764-766.
  14. Beiträge zur Theorie der konvexen Körper. Comptes Rendus du Congrès International des Mathématiciens, Oslo 1936, Bind 2. A. W. Brøgger, Oslo 1937, p. 163.
  15. Om Bevægelser, der er næstenperiodiske paanær Flytninger. Matematisk Tidsskrift, Afdeling B (1937), pp. 75-80.
  16. Verallgemeinerungen einiger Sätze aus der Geometrie der Zahlen. Acta Arithmetica 2 (1937), pp. 230-241.
  17. With B. Jessen: Mengenfunktionen und konvexe Körper. Det Kongelige Danske Videnskabernes Selskab, Mathematisk-Fysiske Meddelelser no. 3, 16 (1938), 31 pp.
  18. Über die neuere Entwicklung der Brunn-Minkowskischen Theorie der konvexen Körper. Neuvième Congrès des Mathématiciens Scandinaves, Helsingfors 1938. Mercators Tryckeri, Helsingfors 1939, pp. 249-272.
  19. On total curvatures of Riemannian manifolds: I. The Journal of the London Mathematical Society 15 (1940), pp. 15-22. (Photocopied copy).
  20. Nepers Regel for den retvinklede sfæriske Trekant. Matematisk Tidsskrift, Afdeling A (1940), pp. 73-77.
  21. Om Udvidelser af Rolles Sætning. Matematisk Tidsskrift, Afdeling A (1941), pp. 1-11.
  22. Om det projektivgeometriske Grundlag for den ikke-euklidiske Trigonometri. Matematisk Tidsskrift, Afdeling B (1941), pp. 18-30. (2 copies of which one contains corrections).
  23. En elementær mekanisk Realisering af Levi-Civitas infinitesimale Parallelforskydning paa Flader. Matematisk Tidsskrift, Afdeling B (1945), pp. 110-112.
  24. On closed surfaces with constant sign of curvature in the projective space. Den 10. Skandinaviske Matematikerkongres i København 26.-30. August 1946. Jul. Gjellerups Forlag, København 1947, pp. 207-212.

Folder: Reprints of articles 1948-83 etc.
There are reprints of the following articles (if there are more than one copy, it is mentioned):
  1. With J. Nielsen: On discontinuous groups of isometric transformations of the non-Euclidian plane. Studies and Essays presented to R. Courant on his 60th birthday, January 8, 1948. Interscience Publishers, Inc., New York 1948, pp. 117-128. (2 copies of which one is photocopied and the other contains corrections).
  2. Estensioni di gruppi discontinui e trasformazioni periodiche delle superficie. Rendiconti della R. Accademia nazionale dei Lincei, Classe di Scienze fisiche, matematiche e naturali (8), 5 (1948), pp. 326-329.
  3. En kongruenssætning for konvekse polyedre. Matematisk Tidsskrift, Afdeling A (1949), pp. 35-43.
  4. On conjugate convex functions. Canadian Journal of Mathematics 1 (1949), pp. 73-77. (3 copies of which there are corrections in two and the third is a photocopy).
  5. Bemærkninger om endelige grupper af afbildningsklasser. Matematisk Tidsskrift, Afdeling B (1950), pp. 90-95.
  6. On the differential geometry of closed space curves. Bulletin of the American Mathematical Society (1), 57 (1951), pp. 44-54.
  7. On Th. Bang's solution of the plank problem. Matematisk Tidsskrift, Afdeling B (1951), pp. 49-51.
  8. A generalization of spherical trigonometry. Den 11te Skandinaviske Matematikerkongres i Trondheim 22-25 August 1949. Johan Grundt Tanums Forlag, Oslo 1952, pp. 139-147.
  9. A remark on convex sets and polarity. Meddelanden från Lunds Universitets Matematiska Seminarium, Supplementband tillägnat Marcel Riesz C. W. K. Gleerup, Lund 1952, pp. 82-89.
  10. On curvature and Levi-Civita's parallelism in Riemannian manifolds. Convegno Internazionale di Geometria Differenziale, Italia, 1953. Edizioni Cremonese, Roma 1954, pp. 99-103.
  11. On the convex hull of point sets with symmetry properties. Rendiconti di Matematica e delle sue applicazioni (5), 14 (1955), pp. 347-350.
  12. Sur les variétés localement convexes des espaces projectifs. Colloque sur les Questions de Réalité en Géométrie, Liège 1955. Georges Thone, Liège; Masson & Cie, Paris 1956, pp. 95-104.
  13. Über konvexe Funktionen mit vorgeschriebenen Niveaumannigfaltigkeiten. Mathematische Zeitschrift 63 (1956), pp. 496-506.
  14. Bemerkungen zur allgemeinen Theorie der diskontinuierlichen Transformationsgruppen. Treizième Congrès des Mathématiciens Scandinaves, Helsingfors 1957. Mercators Tryckeri, Helsingfors 1958, pp. 77-85. (2 copies of which one has corrections).
  15. Om indførelsen af eksponentialfunktionen. Nordisk Matematisk Tidskrift 6 (1958), pp. 109-113, 136.
  16. Jakob Nielsen in memoriam. Nordisk Matematisk Tidskrift 8 (1960), pp. 5-10, 63.
  17. Jakob Nielsen. Festskrift udgivet af Københavns Universitet i anledning af universitetets årsfest, november 1960. Bianco Lunos Bogtrykkeri A/S, København 1960, pp. 99-103.
  18. Jakob Nielsen in memoriam. Acta Mathematica 103 (1960), pp. vii-xix.
  19. A beginners' course in analytic geometry and linear algebra. Report of the Second Conference on Mathematical Education in South Asia. Tata Institute of Fundamental Research, Bombay 1960, pp. 49-56.
  20. Konveksitet gennem tiderne. Dansk Matematisk Forening 1923-1973, København 1973, pp. 103-116.
  21. Om matematikkens begreber og metoder. Det Kongelige Danske Videnskabernes Selskabs Pjeceserie Grundvidenskaben i Dag, 17. Folkeuniversitet i København, 1979. 25 s.
  22. Erinnerungen aus der Studienzeit. Jahrbuch überblicke Mathematik, Bibliographisches Institut, Mannheim, 1980, pp. 155-166. (1 copy with corrections).
  23. Convexity through the ages. Convexity and its applications. Edited by Peter M. Gruber and Jörg M. Wills. Birkhäuser, Basel-Boston-Stuttgart 1983, pp. 120-130. (Photocopied reprint).

There are reprints of the following book reviews and solutions of problems:

  1. Ludwig Bieberbach, Analytische Geometrie. IV. Jahresbericht der Deutschen Mathematiker-Vereinigung, 2. Abteilung 41 (1931), pp. 24-25.
  2. O. Weinberger, Mathematische Volkswirtschaftslehre. Jahresbericht der Deutschen Mathematiker-Vereinigung, 2. Abteilung 41 (1932), pp. 101-102.
  3. G. Julia, Leçons sur la représentation conforme des aires simplement connexes. VIII. Jahresbericht der Deutschen Mathematiker-Vereinigung, 2. Abteilung 42 (1932), pp. 100-101.
  4. Aufgaben und Lösungen. Jahresbericht der Deutschen Mathematiker-Vereinigung, 2. Abteilung 44 (1934), pp. 82.
  5. Aufgaben und Lösungen. Jahresbericht der Deutschen Mathematiker-Vereinigung, 2. Abteilung 45 (1935), pp. 34-35.
  6. C. Juel, Vorlesungen über projektive Geometrie mit besonderer Berücksichtigung der v. Staudtschen Imaginärtheorie (Grundlehren der math. Wissenschaften, Band XLII), XI+287 S., Berlin, J. Springer, 1934. (It is not clear where this review was printed; 2 copies).

Folder: Encyclopedia articles
There are encyclopedia articles written by Fenchel on the following entries:
Babylonsk Matematik [Babylonian mathematics], 2 pages typewritten in Danish in 2 copies.
Cirkeldelning [Division of the circle], 1 page typewritten in Danish.
Grafisk beräkning [Graphical computation], 1 page typewritten in Danish.
Hamilton, Sir William Rowan, 1 page typewritten in Danish.
Hardy, Godfrey Harold, 1 page typewritten in Danish.
Hilbert, David, 2 pages typewritten in Danish.
Matematik [Mathematics], 14 pages typewritten in Danish in 2 copies.
Mängdlära, mängdteori [Set theory], 3 pages typewritten in Danish.
Neumann, John von, 1 page typewritten in Danish.
Oändelighetsbegreppet [The concept of infinity], 3 pages typewritten in Danish.
Parallellförskjutning [Map made by parallel movement], 1 page typewritten in Danish.
Pi, 2 pages typewritten in Danish.
Projektiv geometri [Projective geometry], 2 pages typewritten in Danish.
Punkt, [Point] 1 page typewritten in Danish.
Pyramid, [Pyramid] 1 page typewritten in Danish.
Reella funktioner [Real functions], 1 page typewritten in Danish.
Rektifikation [Rectification], 1 page typewritten in Danish.
Riemann, Bernhard, 1 page typewritten in Danish.
Rum [Spaces], 2 pages typewritten in Danish.
Stereografisk projektion [Stereographical projection], 1 page typewritten in Danish.
Tidsskrifter [Journals], 3 pages typewritten in Swedish.
Three reprints of Ur Svensk Uppslagsbok with Fenchel's article on "mathematics". Reprints of Salmonsen Leksikon-Tidsskrift issue no. 7 (1943) and issue no. 12 (1951) with articles on Hilbert respectively Harald Bohrwritten by Fenchel.

T. Bonnesen und W. Fenchel: Theorie der Konvexen Körper 1934 (2 copies)
The book was published in the series: Ergebnisse der Mathematik und Ihrer Grenzgebiete, Dritter Band, Verlag von Julius Springer, Berlin 1934. There are 2 copies of which one has been dedicated to Jakob Nielsen by the author and the other is uncut.

Box 10

Folder: Jakob Nielsen manuscripts 1921
A photocopy of 7 pages of handwritten notes to Nielsen's lectures: Die Abbildungstypen geschlossener Flächen und ihre Beziehungen zu unendlichen Gruppen (2 Vorträge, gehalten in Breslau am 9. u. 11.3.21 von J. Nielsen).
A photocopy of a typewritten manuscript in English that is a translation of the above mentioned lectures, because the title is: The mapping types of closed surfaces and their relationship with infinite groups (2 Lectures, given in Breslau on the 9th and 11th of March 1921, by J. Nielsen). It is 13 pages long.

Folder: Jakob Nielsen's collected works
Typewritten manuscripts for sections with comments etc. in Jakob Nielsen: Collected Matematical Papers. It is:
  • Table of Contents.
  • Joan S. Birman: Nielsen's investigation of surface mapping classes. 23+6 pp.
  • Heiner Zieschang: On Nielsen's work on finite mapping classes and related subjects. 7 pp. In two versions both with corrections.
  • Robert F. Brown: Nielsen fixed point theory. 13 pp.
  • Werner Fenchel: Nielsen's contributions to the theory of discontinuous groups of isometries of the hyperbolic plane. In two versions on respectively 35 and 36 pages.
  • Discontinuous groups of motions. 3 typewritten pages with a summary in English of the Danish article "Diskontinuerte Flytningsgrupper", Matematisk Tidsskrift B (1923), pp. 53-61.
  • A photocopy of Mathematical Reviews, May 1948, pp. 221-222, where there is a reference to a work by Nielsen in a review of an article by A. Markov.

Manuscript: Vagn Lundsgaard Hansen: En dansk matematikers eventyr
Typewritten manuscript in Danish entitled The adventure of a Danish mathematician 9 pages long. It appeared in Carlsbergfondets Årsskrift [The year book of the Carlsberg Foundation] 1983.

Obituary of Jakob Nielsen
A reprint of: Werner Fenchel: Jakob Nielsen in memoriam. Acta mathematica 103 (1960), pp. xvii-xix. Included is a typewritten list on 3 pages containing which lectures etc. Nielsen attended in his time of study.

Folder: Jakob Nielsen reprints
There are reprints of the following works by Jakob Nielsen:
  • Kurvennetze auf Flächen. Inaugural-Dissertation, C. Schaidt, Kiel 1913.
  • Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden. Mathematische Annalen 78 (1917), pp. 385-397.
  • Über die Isomorphismen unendlicher Gruppen ohne Relation. Mathematische Annalen 79 (1918), pp. 269-272.
  • Über fixpunktfreie topologische Abbildungen geschlossener Flächen. Mathematische Annalen 81 (1920), pp. 94-96.
  • Die Gruppe der dreidimensionalen Gittertransformationen. Det Kongelige Danske Videnskabernes Selskab, Mathematisk-Fysiske Meddelelser, no. 12, 5 (1924), 29 pp.
  • Zur Topologie der geschlossenen Flächen. Den sjette Skandinaviske Matematikerkongres i København den 31. Aug.- 4. Sept. 1925. Det Hoffensbergske Etablissement, København 1926, pp. 263-276.
  • Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. Acta mathematica 50 (1927), pp. 189-358.
  • Topologie der transformations des surfaces. l'Enseignement mathématique 35 (1936), pp. 269-287.
  • Die Struktur periodischer Transformationen von Flächen. Det Kongelige Danske Videnskabernes Selskab, Mathematisk-Fysiske Meddelelser, no. 1, 15 (1937), 77 pp.
  • En Sætning fra Fladeafbildningernes Topologi. Norsk Matematisk Tidsskrift 23 (1941), pp. 96-99.
  • Abbildungsklassen endlicher Ordnung. Acta mathematica 75 (1942), pp. 23-115. (3 copies).
  • Fixpunktsfrie Afbildninger. Matematisk Tidsskrift B (1942), pp. 25-41.
  • Kommutatorgruppen for det frie produkt af cykliske grupper. Matematisk Tidsskrift B (1948), pp. 49-56.


Box 11

Plastic folder: G. Fubini: Introduzione alla teoria dei gruppé discontinué e delle funzioni automorfe 1908
The handwritten remarks of Fenchel in Italian to the article G. Fubini: Introduzione alla teoria dei gruppé discontinué e delle funzioni automorfe Pisa 1908. The remarks are 17 pages long.

Folder: Miscellaneous
Miscellaneous notes, references and other material.