**Max Dehn** wrote one of the first systematic expositions of topology (1907) and later formulated important problems on group presentations, namely the word problem and the isomorphism problem.

Let us begin by giving some details of Max Dehn's family. His father, Maximillian Moses Dehn (15 March 1841 - 16 April 1897), was a medical doctor. Maximillian was one of a large family having nine siblings: Arnold, Henne Hannah, Gustav, Rudolf Joseph, Otto Carl Isaak, Hannah Bertha, Martin, Elizabeth Charlotte, and Moses Wilhelm. Maximillian married Berta Raf Hirsch (15 April 1845 - 2 February 1926) in Hamburg, Germany, on 2 May 1867. They had nine children: Rudolph Bernhard, Elizabeth Hanne Adeline, Henriette Marianne, Max Wilhelm (the subject of this biography), Berta Meta, Karl Arnold, George Ernst, Charlotte, and Marie Auguste. The family was racially Jewish but did not think of themselves as Jews, rather they were Germans. Max was brought up in Hamburg where he attended the Wilhelm Gymnasium. This school, founded in 1881, was named after emperor Wilhelm I and when Dehn attended the school it was situated on Moorweidenstrasse. This school prepared students for university and, after graduating, Dehn entered the University of Freiburg.

At Freiburg, Dehn studied mathematics but, as was common for students at this time, he did not spend his whole time at one university. He also studied at Göttingen where, under David Hilbert's supervision, he obtained his doctorate in 1900 for a thesis entitled *Die Legendreschen Sätze über die Winkelsumme im Dreieck* Ⓣ. In this thesis he proved the Saccheri-Legendre theorem which states that in absolute geometry the sum of the angles in a triangle is at most 180°. By absolute geometry, we mean geometry satisfying the axioms of Euclidean geometry except for the parallel postulate. In August 1900, Hilbert gave his famous address at the International Congresses of Mathematicians held in Paris. In it he presented ten problems which he believed were important for the development of mathematics. In fact, the published version of his talk contains 23 such problems. In 1900 Dehn solved the 3rd of these problems namely:

Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?

Dehn showed that the answer to this question is "no", constructing a counterexample using what is today called the 'Dehn invariant'. This was the first of Hilbert's problems to be solved. He submitted his paper solving the 3rd Hilbert Problem to the University of Münster as his Habilitation thesis before the end of 1900 and, in the following year, he was appointed as a Privatdocent at Münster. He held this position until 1911. In 1907, jointly with Poul Heegaard, Dehn wrote one of the first systematic expositions of topology for the *Enzyklopädie der mathematischen Wissenschaften* [2]:-

It offers the first systematic representation of a research direction of great depth and beauty - going back to Euler, Gauss, Listing, Riemann, Möbius, Betti and Poincaré - which was then called 'analysis situs' and today probably be called geometric topology.

Dehn next attempted to solve the Poincaré conjecture but, of course, he failed. However, his attempts led him to publish the paper *Über die Topologie des dreidimensionalen Raumes* Ⓣ (1910) in which he discussed 3-dimensional manifolds with trivial homology, constructed from the complement of the trefoil knot and various other knots. In order to study such manifolds, he constructed the Cayley graph of their fundamental groups. In 1911 Dehn was appointed as an extraordinary professor at Kiel, and from 1913 until 1921 he was a full professor at the University of Breslau. He married Antonie Landau on 23 August 1912; they had three children Helmut (1914-2007), Maria (1915-2013), Eva Agathe (1919-2008) all born in Breslau.

In 1914 Dehn published the paper *Die beiden Kleeblattschlingen* Ⓣ. Joan Birman describes this paper in [5]:-

Written in1914, not long after the discovery of the fundamental group of a topological space, it tackles a simple and beautiful problem: to confirm a property of the simplest knot which is suggested by five minutes of experimentation: that the right and left trefoil knots are not isotopic. The fundamental group, very new in1914, was used by Dehn in a sophisticated fashion, via the investigation of its outer automorphism group. Having used it to prove the trefoil is not amphicheiral, Dehn's paper concludes with a brief investigation of the outer automorphism group of the fundamental group of the figure eight knot, which is amphicheiral. He identifies(as was shown later)its generators. ... before1984we really didn't have any simple tests for non-amphicheirality, so that Dehn's work(which at first glance looks like the use of a cannon to kill a sparrow)remained central to the subject for nearly70years.

Of course, 1914 marks the beginning of World War I and Dehn's career was interrupted while he served in the army from 1915 to 1918. After military service, he returned to his position as professor at Breslau, but in 1921 he was appointed to the chair of Pure and Applied Mathematics at the University of Frankfurt, succeeding Ludwig Bieberbach. There he organised a famous seminar on the history of mathematics. André Weil writes [4]:-

A humanistic mathematician who saw mathematics as one chapter - certainly not the least important - in the history of human thought, Dehn could not fail to make an original contribution to the historical study of mathematics, and to involve his colleagues and students in the project. This contribution, or rather this creation, was the historical seminar of the Frankfurt mathematics institute. Nothing could have seemed simpler or less pretentious. A text would be chosen and read in the original, with an effort to follow closely not only the superficial lines but also the thrust of the underlying ideas. ... It was only later that I attended it, on subsequent visits to Frankfurt, a place I made a point of visiting as often as I could. I am not sure whether it was already in the summer semester of1926that, during a seminar session devoted to Cavalieri, Dehn showed how this text had to be read from the viewpoint of the author, taking into account both what was commonly accepted in his lifetime and the new ideas that Cavalieri was trying the best of his ability to implement. Everyone participated in the discussion, contributing what he could to the group effort.

In 1932 Dehn wrote the essay *Das Mathematische im Menschen *(Mathematical ability in humans) which he published in *Scientia*, an Italian journal produced in Bologna by Federigo Enriques [2]:-

In this essay, Dehn attempted, as he wrote "to link the science of mathematics with the whole of human activity". According to Dehn, only a few preconditions are necessary to bring about mathematics. Mathematics, he claimed, arises of its own accord whenever mathematical emotion and logical conclusions come together - and all human beings have both capacities. Dehn thus positioned "mathematical ability" at a basic anthropological level. The belief that mathematics is deeply rooted in human nature, in much the same way that music is based on innate musical abilities, gave him the confidence that non-professional mathematicians could also be made to appreciate this science.

See the first few paragraphs of *Dehn on Mathematical abilities* at THIS LINK.

On 30 January 1933 Hitler came to power in Germany and on 1 April there was the so-called "boycott day" when Jewish shops were boycotted and Jewish lecturers were not allowed to enter the university. On 7 April 1933 the Civil Service Law was passed which provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired but there were exemptions for those who had fought for Germany in World War I. Dehn qualified under this exemption clause so continued to head the mathematics seminar at University of Frankfurt. However, decisions at the Nuremberg party congress in the autumn of 1935 made it clear that non-Aryans would no longer be able to keep their posts even if they had served in World War I. Dehn was forced to resign but remained in Frankfurt. In 1936 he sent his children out of Germany. His son Helmut went to the United States and his daughters Maria and Eva were sent to England to continue their education. Dehn lectured in a number of European countries over the next couple of years and spent from January to April 1938 in England with his daughters. He continued to publish articles. In 1936, he published *Raum, Zeit, Zahl bei Aristoteles vom mathematischen Standpunkt aus* Ⓣ in *Scientia*.

See the first few paragraphs of *Max Dehn on Space, Time and Number in Aristotle* at THIS LINK

In 1938 he published *Die Gruppe der Abbildungsklassen* Ⓣ which contains the important ideas known today as a 'Dehn twist'. Dehn and his wife were in Frankfurt on Kristallnacht (so called because of the broken glass in the streets on the following morning), the 9-10 November 1938. Many Jews were murdered, hundreds were seriously injured, and thousands were subjected to horrifying experiences. Thousands of Jewish businesses were burnt down together with over 150 synagogues. The Gestapo arrested 30,000 well-off Jews and a condition of their release was that they emigrate. On 11 November 1938 Dehn was arrested but, with so many Jews being held, the prisons were full and he was released later on the same day. Immediately Dehn and his wife, fearing re-arrest, fled to the home of Willi Hartner, friends in Bad Homburg north of Frankfurt. Hartner wrote [11]:-

Unforgettable for those who saw him at the time was his calmness, his philosophical composure. For the conversations centred not on the events of the day, but on the relationship of mathematics to art, on problems of archaeology, and finally on the concept of humanity of Confucius.

After a few weeks the Gestapo became less active in their persecution of Jews and Dehn and his wife risked fleeing to Hamburg where they lived for a while at the home of one of Dehn's elder sisters before escaping, first to Denmark in January 1939 and from there to Norway. Dehn was soon given a temporary position at he Technische Hochschule in Trondheim as a replacement for Viggo Brun (1885-1978) who was on leave. However, on 1 March 1940 Germany invaded Norway and Trondheim fell to the Germans on 9 April. Dehn fled the city but, despite the great risks, was back in the city by June and began to plan his move to the United States.

In October 1940 the Dehns emigrated to the USA, travelling from Norway to Stockholm in Sweden, then on to Moscow in Russia from where they took the trans-Siberian train to Vladivostok. From there they took a ship to Kobe in Japan and then another ship to San Francisco where they arrived on 1 January 1941. Once in the USA Dehn taught at several universities and colleges, for instance at the University of Idaho in Pocatello, the Illinois Institute of Technology and St John's College in Annapolis, Maryland. Paul L Chessin has described some interesting personal recollections of Dehn while at Madison, Wisconsin:-

In1945, [Dehn]replaced Rudolph Langer, then head of the mathematics department in Madison, Wisconsin, who went on sabbatical to the University of Texas. Prof Dehn came to teach several graduate courses. I attended his course in 'Non-linear Partial Differential Equations'. We were delighted to have a German-speaking instructor since we were to take our German language examinations(required for the doctorate). He, in turn, refused since it was that term that he was to take his examination for US citizenship! Max would hold forth in the Rathskellar(the only campus beer establishment within the Big Ten Universities). We learned more in the beer stube - such as his personal life. He was the black sheep in his family. So long as he remained in his university, he was supported. Thus he received many doctorates. He would declaim in Greek, some passages from the classics, beer stein in hand. At the time for setting out final grades, he merely called the attendance and one by one asked essentially for the titles of the chapters in the textbook. Clearly we could glance ahead to be assured of the correct answer. I believe he gave something like18A's and3-4B's. Throughout the class sessions he would interrupt with some voiced concerns about passing his citizenship examination.

However Dehn was unable to find a full-time position and Saunders Mac Lane recently wrote:-

... most mathematicians fleeing Europe were helped to some sort of position in the United States ... I recall two cases of failures: Max Dehn, noted for work in topology, got only a weak position.

The weak position, referred to by Mac Lane, was at Black Mountain College. This college had no accredited degrees and taught mainly creative arts. There was no trained mathematician on the staff when Dehn was invited to give two lectures there in 1944. He realised that he could not lecture on advanced mathematics so he gave his lectures on 'Common roots of mathematics and ornamentics' and 'Some moments in the development of mathematical ideas'. He was offered a permanent post there at $25 a month. He held out for $40 a month which was agreed. Dehn joined the Faculty in 1945 and remained there until his death. Dehn was the only mathematician ever to teach at the College which closed in 1956. David Peifer writes [14]:-

Dehn taught Mathematics, Philosophy, Greek, and Italian. In Mathematics his courses included History of Mathematics and Projective Geometry. He is remembered for his love of nature and the arts. His lectures frequently included tangents on philosophy, the arts, and their connections to mathematics. He was best when lecturing in a socratic style. He was fond of giving these lectures while hiking with his students through the woods. He is remembered as a caring and devoted teacher.

Dehn's outstanding research record is in stark contrast with the low level of his final post. He was an intuitive geometer, stimulated by Hilbert's axiomatic approach to the subject. His work in topology had led him into the study of groups, particularly group presentations which arise naturally from topological considerations. In his 1911 paper *Über unendliche diskontinuierliche Gruppen* Ⓣ, Dehn formulated important problems on group presentations, namely the word problem and the isomorphism problem. The word problem asks the fundamental question of whether there is an algorithm to determine whether a word in a group given by a presentation is trivial. It has since been shown that no such algorithm exists in general. Research on questions of this type are still of major importance in combinatorial group theory. Dehn also wrote on statics, projective planes and, as we have noted above, on the history of mathematics.

After World War II ended, the German Mathematical Society (DMV) was re-established in 1948 and Erich Kamke (1890-1961) wrote to all those who had been expelled asking them to rejoin. Dehn refused and we give his reply, written in August 1948, in which he states clearly his reasons (see for example [2]):-

I bear no grudge of any kind. As you may be aware, I am once again in close contact with several mathematicians in Germany, of course primarily with those whom I was particularly close to. But I cannot rejoin the German Mathematical Society. I have lost the confidence that such an association would act differently in the future than it did in1935. I fear it would, once again, not resist an unjust measure coming from outside. The DMV did not have the custody of very important values. My negative impression is caused by the fact that it did not dissolve itself in1935and that not even a considerable number of mathematicians left the association. I am not afraid that the DMV will once again expel Jews, but perhaps next time it will be so-called communists, anarchists or "coloured people". Contact with Germany, especially with German mathematicians, is very dear to my heart.

Paul L Chessin has described some interesting personal recollections of Dehn:-

In1945, [Dehn]replaced Rudolph Langer, then head of the mathematics department in Madison, Wisconsin, who went on sabbatical to the University of Texas. Prof Dehn came to teach several graduate courses. I attended his course in Non-linear Partial Differential Equations.

We were delighted to have a German-speaking instructor since we were to take our German language examinations (required for the doctorate). He, in turn, refused since it was that term that he was to take his examination for US citizenship !

Max would hold forth in the Rathskellar (the only campus beer establishment within the Big Ten Universities). We learned more in the beer stube - such as his personal life. He was the black sheep in his family. So long as he remained in his university, he was supported. Thus he received many doctorates.

He would declaim in Greek, some passages from the classics, beer stein in hand. At the time for setting out final grades, he merely called the attendance and one by one asked essentially for the titles of the chapters in the textbook. Clearly we could glance ahead to be assured of the correct answer. I believe he gave something like18A's and3-4B's.

Throughout the class sessions he would interrupt with some voiced concerns about passing his citizenship examination.

Dehn retired at the end of the academic year 1951-1952 and was made Professor Emeritus. He continued to work at Black Mountain College, giving advice to staff and students. Shortly after supervising the removal of a number of dogwood trees from the college campus, he became ill and died of an embolism. He is buried in the woods that he loved on the Black Mountain College campus.

**Article by:** *J J O'Connor* and *E F Robertson*

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