# Emanuel Sperner

### Quick Info

Born
9 December 1905
Waltdorf, Upper Silesia (now Poland)
Died
31 January 1980

Summary
Emanuel Sperner was a German mathematician who worked in the foundations of geometry.

### Biography

Emanuel Sperner's father was an estate agent who worked in Neisse (now Nysa), the main town in the region, which was known as the "Silesian Rome" because of its many old churches. Waltdorf was a village close to Neisse. His name, Emanuel, was an Old Testament name meaning "God with us" and Bachmann writes in [1] that Sperner was proud of his biblical name and felt a closeness with Emanuel Kant who he greatly admired. Sperner attended the Carolinum Gymnasium in Neisse where he received an excellent education. As well as giving him an excellent background in mathematics he graduated from the school in 1925 having learnt six languages. In particular, he always spoke of his debt to his German teacher G Janocha who taught him to think in a logical and clear way.

In 1925, the year he graduated from the Carolinum Gymnasium in Neisse, Sperner entered the University of Freiburg. However life did not prove that easy for the young student who had to overcome illness. As was the custom for students at this time, Sperner did not spend the whole of his university career at a single university but, after two semesters, he went to the University of Hamburg. At Hamburg he had Wilhelm Blaschke as his thesis advisor but he was also taught and advised by Otto Schreier. Sperner greatly appreciated Blaschke's help and often in later life spoke about how much he owed to his advice. Sperner was awarded his doctorate, with distinction, on 15 November 1928 for his thesis Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes . This thesis contained the important result which today is known as Sperner's lemma. Sperner published a paper with the same title as his thesis, which he submitted to the Hamburg Mathematical Seminar in June 1928 and it was published later that year. In the paper he writes:-
The suggestion for dealing with these questions was given to me by Otto Schreier in Hamburg, and I would like to express my warmest thanks to him at this point.
The editors of [1] describe the contents:-
Sperner was just 22 when in 1928 he published his second paper, 'Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes' (1928), in which he developed new methods, with which he was able to prove the Lebesgue Covering Theorem, the invariance of dimension and also the invariance of domains, in an elementary way. The basic 'Sperner Lemma' used for this created great interest in topology: B Knaster, C Kuratowski, and S Mazurkiewicz were able to prove the Brouwer Fixed Point Theorem with its help. Sperner himself exerted great influence by applying the ideas he developed too.
We should give an indication of what 'Sperner's Lemma' states. First we define a Sperner labelling:
Definition. A Sperner labelling of a triangle's triangulation is a labelling that satisfies the following conditions:
1. The vertices of the original triangle are labelled with three different letters $A, B$, and $C$.
2. Vertices of the triangulation that lie on the side $AB$ are labelled either $A$ or $B$. A similar condition holds for the vertices on sides $BC$ and $AC$.
3. There is no restriction on labelling of the vertices of the triangulation inside the original triangle.

Then Sperner's Lemma states:
Every Sperner labelling contains at least one triangle of the triangulation labelled ABC.
As the above quote by the editors of [1] indicates, this paper was not the first that Sperner published. Note zu der Arbeit von Herrn B L van der Waerden: "Ein Satz über Klasseneinteilungen von endlichen Mengen" was submitted to the Hamburg Mathematical Seminar in January 1927 and published later that year. In February 1927 he submitted his paper Ein Satz über Untermengen einer endlichen Menge to Mathematische Zeitschrift and it was published early in 1928. Now this last mentioned paper contains the result which today is known as Sperner's Theorem. This theorem describes the largest possible collection of subsets of a given set with the property that no subset in the collection contains any other subset in the collection. The fact that Sperner had discovered two results while still a research student which remain of major significance today is quite remarkable. But he had other results of great significance, for example his paper Über einen kombinatorischen Satz von Macaulay und seine Anwendungen auf die Theorie der polynomideale (1929). The importance of this paper can be seen from the fact that an English translation On a combinatorial theorem of Macaulay and its applications to the theory of polynomial ideals was published in 2008. Marco Fontana, reviewing the 2008 translation, writes:-
This is the English translation of an interesting and still relevant paper published in 1930 [actually the paper was submitted in July 1929 and published later that year]. The aim of this paper was to produce a simpler and shorter proof of a combinatorial theorem proved by F S Macaulay in 1927. The author also added new versions of some of the most relevant applications of this theorem to the theory of polynomial ideals and the Hilbert characteristic function.
In winter 1929-30 Sperner gave his first lecture course 'Analytic geometry and algebra II' at the University of Hamburg. Otto Schreier, who was only 28 years old, died in June 1929. He had planned to publish his lectures as a book but had not made any progress on that project before his death. Sperner now took over the project, and decided to publish Schreier's lectures with his own contributions added. "Einführung in die analytische Geometrie und Algebra. I" appeared in print in 1931. Henry George Forder begins his review [4]:-
Schreier intended eventually to publish his Hamburg lectures on analytical geometry and algebra in book form; and after his premature death in 1929 these were completed and edited by his pupil Sperner. Three-quarters of the present first volume discuss the analytical geometry of $n$ dimensions and the relevant algebra in a unified way; no special knowledge is assumed, but considerable demands are often made on the attention of the reader. The first chapter deals with affine space and linear equations. ... Chapter two, [is] on Euclidean space and determinant theory ... The second part, [is] on the theory of fields ...
Sperner habilitated at Hamburg in the summer of 1932 after submitting his major work Über die fixpunktfreien Abbildungen der ebene . At this stage Sperner was only 26 years old and considered too young to be taken seriously as a university professor so, in August 1932 immediately after habilitating, he travelled to Peking in China taking a route via North America and Japan. He went at the invitation of the China Foundation for the Promotion of Education and Culture and, arriving in September 1932, he became a visiting professor at the National University of Peking. He taught in Peking until 1934, having some outstanding students such as Shiing-shen Chern and Ky Fan (1914-2010). In 1934 he married Annemarie Voss, who was a student at the University of Hamburg. He was appointed as an ordinary professor at the University of Königsberg on 1 November 1934. The position had become vacant since Kurt Reidemeister who had been appointed to the chair in 1927 had been forced out by the Nazis in 1933 as "politically unsound". In 1935 Sperner published "Einführung in die Analytische Geometrie und Algebra II". Henry George Forder writes in [5]:-
This final volume contains an introduction to the theory of (finite) matrices, which has already been published separately and is now out of print, together with a treatment of $n$-dimensional analytic geometry from the projective, affine and metric points of view, as far as the classification of quadrics.
Before continuing to describe Sperner's career, let us move forward to the English translations of these books. Introduction to Modern Algebra and Matrix Theory by Schreier and Sperner was published in 1951. Kurt Hirsch writes [6]:-
'Einführung in die Algebra und Analytische Geometrie' has for many years been a favourite text-book for first year undergraduates in German universities. One of its unique features is the simultaneous treatment of fundamental concepts in algebra and in affine and projective geometry. The present skilful English translation has incorporated the "Vorlesungen über Matrizen" (as in the German second edition), but has omitted the long final chapter on projective geometry in $n$ dimensions. The result is that the accent is much more on the algebraic developments than in the original and that the geometry serves more as illustration than as motivation. This is also expressed in the changed title from which the reference to geometry has disappeared. ... Apart from this change of emphasis the book has preserved its carefully didactic character. The thorough explanations of concepts, the meticulous statements of assumptions, the elegance of treatment (which is not always brief, but always lucid) deserve the highest praise.
T C Holyoke, also reviewing this English translation, writes:-
This outstanding book forms a good introduction to some of the topics and methods of modern algebra, and appears to be well suited for use as a text in a course of that nature. The treatment of the subject matter is concrete, and frequent geometric illustrations and motivations are given.
In 1961 the omitted material on projective geometry was translated into English and published as Projective Geometry of n Dimensions. J W Archbold writes [2]:-
The reviewer feels that it is a great pity, as regards the teaching of projective geometry, that the present book has taken so long to appear. Indeed it arrives at a period when the projective geometry in undergraduate courses is being whittled down in favour of vector space theory. If it had appeared sooner this book would have done much to make projective geometry more attractive.
We return to our description of Sperner's life and career. In 1940, while in Königsberg, his wife died from a blood disorder following the birth of their daughter. Now Friedrich Bachmann was appointed to lecture at the University of Königsberg in the summer of 1941. He writes in [3] about the advice he received from Sperner:-
Sperner, who had a natural teaching talent himself, advised me to keep giving easy-to-understand lectures, but to increase the difficulty of the exercises and he handed me a collection sufficiently hard problems.
Of course, by this time World War II was disrupting normal life throughout Europe. From the spring of 1942, Sperner worked as an assistant in the Navy's Weather Service. He married his second wife, Antonie Schwörer, in Biberach an der Riss; their two sons Emanuel Sperner Jr and Peter Sperner both became mathematicians. While still undertaking this war work in 1943, Sperner accepted an appointment as a professor at Strassburg. He began teaching there but in the autumn of 1944 Allied troops approached the city and the university was closed. A request was submitted to transfer Sperner to the University of Hamburg where, not only were there staff shortages, but he had contacts with the naval observatory there though his war work. However, he was released from his war work with the Navy and assigned to undertake work at the Mathematics Institute at Oberwolfach. Wilhelm Süss was the director of the Oberwolfach Institute and Sperner was appointed as deputy director. Together with Süss, he must be considered as one of the founders of the Oberwolfach Mathematics Research Institute. In 1946 Sperner became a guest professor at the University of Freiburg but continued to work at Oberwolfach. The war years had disrupted Sperner's publication record and the first paper he published following the war was Die Ordnungsfunktionen einer Geometrie (1948). In the paper, Sperner, who gives his address as Oberwolfach, writes:-
I lectured on this subject for the first time in November 1943 in Bucharest and Timișoara.
Leonard M Blumenthal, reviewing this paper, writes:-
The author introduces the concept of an order-function for the purpose of making geometric order relations amenable to algebraic formulation and treatment.
This short paper only announced Sperner's ideas and in the following year he published a detailed account of his concept of an order-function, showing which betweenness and separation properties may be algebraically described.

In 1949 Sperner was appointed as an ordinary professor at the University of Bonn. While at Bonn, he developed a type of generalised affine space, influenced by Hilbert's 'Foundations of geometry', which was taken up by other mathematicians, particularly those in Italy. He remained at Bonn for five years before being appointed as an ordinary professor at the University of Hamburg in 1954. He spent the rest of his career at Hamburg, being rector of the university from 1963 to 1965. He retired in 1974 and was made professor emeritus at that time. While holding the chair at Hamburg, Sperner spent several periods as a visiting professor in a number of different institutions. He spent the year 1961-62 at the University of Pittsburgh in the United States, he was at the University of South Africa in Pretoria in 1966, at the University of Witwatersrand in Johannesburg in 1969, at the University of California at Berkeley in the United States in 1970 and in the same year he visited the University of Sao Paulo in Brazil.

Sperner had a long association with the German Mathematical Society (DMV). He joined the Society in 1930 and in 1935 he became secretary following Ludwig Bieberbach's resignation from that position. Bieberbach had published an "open letter" in the Jahresbericht, the journal of the Society, which was highly critical of Harald Bohr because he had attacked Bieberbach's racist views. Towards the end of 1935 Sperner gave up the position of secretary to become editor of the Jahresbericht, replacing Konrad Knopp who had resigned. Sperner continued as editor until the closure of the Jahresbericht at the end of 1943. He also took on the role of deputy treasurer, taking over the duties of Helmut Hasse, the treasurer, when he was on military service. He was a member of the board which, in 1938, wrote to Jewish members asking them to resign. He write to the other three members of the board (Helmut Hasse, Conrad Müller and Wilhelm Süss) on 28 March 1939 listing all remaining Jewish mathematicians who were still being sent reports of meetings of the DMV. Sperner remained a member of the German Mathematical Society until his death in 1980.

Sperner was, the editors of [1] note, one of the leading geometers of his day. They write:-
His group-theoretical proof of Desargues' Theorem remains a lasting, and much admired, result in absolute axiomatics; his theories of ordering functions and of weakly affine spaces are secure pieces of the discipline of the foundations of geometry.
Sperner received much recognition for his contributions. He was elected to membership of the Königsberger Gelehrten Gesellschaft, the Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, the Rheinisch-Westfälischen Akademie der Wissenschaften, and the Joachim Jungius Society of Sciences in Hamburg. In 1958 he was awarded a Medal from the University of Helsinki, in 1973 he was elected an honorary member of the Hamburg Mathematical Society, and in 1975 received an Honorary Doctorate from the Mathematics Department of the Free University of Berlin. In 1978 a colloquium and celebration was held to make the 50th anniversary of his doctorate.

The Black Forest had always been an area that held a special attraction for Sperner and, after he retired, he built a home in Sulzburg-Laufen, near Badenweiler. He lived there until his death in 1980.

### References (show)

1. W Benz, H Karzel and A Kreuzer (eds.), Emanuel Sperner: gesammelte Werke (Heldermann Verlag, Lemgo, 2005).
2. J W Archbold, Review: Projective Geometry of n Dimensions, by Otto Schreier and Emanuel Sperner, Biometrika 49 (3-4) (1962), 586.
3. F Bachmann, Emanuel Sperner: in memoriam, Jahresber, Deutsch. Math.-Verein. 84 (1) (1982), 45-55.
4. H G Forder, Review: Einführung in die analytische Geometrie und Algebra. I, by O Schreier and E Sperner, The Mathematical Gazette 17 (222) (1933), 57-58.
5. H G Forder, Review: Einführung in die analytische Geometrie und Algebra. II, by O Schreier and E Sperner, The Mathematical Gazette 21 (244) (1937), 240-241.
6. K A Hirsch, Review: Introduction to Modern Algebra and Matrix Theory, by O Schreier and E Sperner, The Mathematical Gazette 37 (321) (1953), 228-229.
7. T C Holyoke, Review: Introduction to Modern Algebra and Matrix Theory, by O Schreier and E Sperner, The Mathematics Teacher 45 (6) (1952), 483.
8. H Karzel, Emanuel Sperner: Leben und Werk, Mitt. Math. Ges. Hamburg 25 (2006), 23-32.
9. H Karzel, Emanuel Sperner: Begründer einer neuen Ordnungstheorie, Mitt. Math. Ges. Hamburg 25 (2006), 33-44.
10. H Karzel, Erinnerungen an Emanuel Sperner aus den Jahren 1948-1968 und Emanuel Sperners Beiträge zur metrischen Geometrie und ihre Bedeutung für die Entwicklung der Geometrie, Mitt. Math. Ges. Hamburg 11 (2) (1983), 217-231.
11. M S Knebelman, Review: Zur Begrundung der Geometrie im Begrenzten Ebenstuck, by Emanuel Sperner, Amer. Math. Monthly 46 (4) (1939), 230.
12. A J Knight, Review: Projective Geometry of n Dimensions, by Otto Schreier and Emanuel Sperner, The Mathematical Gazette 46 (355) (1962), 83.
13. H Lenz, Beiträge Emanuel Sperners zur Vereinfachung komplizierter Mathematik, Mitt. Math. Ges. Hamburg 11 (2) (1983), 196-203.
14. E M Schröder, Emanuel Sperners Forschungsbeiträge zur Spiegelungsgeometrie, Mitt. Math. Ges. Hamburg 25 (2006), 45-56.
15. H Zeitler, Der Didaktiker Emanuel Sperner, Mitt. Math. Ges. Hamburg 25 (2006), 5-13.
16. H Zeitler, Emanuel Sperner und die Schulmathematik, Praxis Math. 23 (8) (1981), 237-243.
17. Zum Gedenken an Emanuel Sperner, Resultate Math. 4 (2) (1981), 125-127