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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.Dehn studied at Göttingen under Hilbert's supervision obtaining his doctorate in 1900. From 1921 until 1935 he held the chair of Pure and Applied Mathematics at the University of Frankfurt but he was forced to leave his post by the Nazi regime in 1938.
In 1940 he emigrated to the USA, travelling there via Scandinavia, Russia and Japan. Once in the USA he taught at several universities and colleges, for instance the University of Idaho, the Illinois Institute of Technology and St John's College in Annapolis, Maryland. However he was unable to find a full-time position and Saunders MacLane 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 MacLane, 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.
Dehn's outstanding research record is in stark contrast with his low level final post. He was an intuitive geometer, stimulated by Hilbert's axiomatic approach to the subject. Dehn had solved the third of Hilbert's 23 problems on the congruence of polyhedra. In 1907 Dehn wrote one of the first systematic expositions of topology jointly with Heegaard. At that time topology was called 'analysis situs'.
Dehn's work in topology had led him into the study of groups, particularly group presentations which arise naturally from topological considerations. 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 the history of mathematics.
Paul L Chessin has described some interesting personal recollections of Dehn:-
In 1945, [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.References (6 books/articles)
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 10 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 like 18 A's and 3-4 B's.
Throughout the class sessions he would interrupt with some voiced concerns about passing his citizenship examination.
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