Max Dehn

Max Wilhelm Dehn (born November 13, 1878 in Hamburg , † June 27, 1952 in Black Mountain , North Carolina ) was a German-American mathematician . He was the first to solve one (the third) of Hilbert's 23 math problems .

Life

Max Dehn was born into a Jewish family in Hamburg and graduated from the Wilhelm-Gymnasium in Hamburg in 1896 . Dehn then studied at the Albert Ludwig University of Freiburg and the Georg August University of Göttingen . He then received his doctorate in Göttingen with David Hilbert with the dissertation The Legendresche theorems on the sum of angles in the triangle in 1900. In 1901 he completed his habilitation at what was then the "Academic Training Institute" in Münster , where he was a private lecturer at the Westphalian Wilhelms University until 1911 . In this habilitation thesis he was the first to solve one (the third) of Hilbert's 23 mathematical problems , but his presentation was not very transparent and complicated and was simplified and completed by Weniamin Kagan and Hugo Hadwiger . From 1911 he was an associate professor at the Christian Albrechts University in Kiel and from 1913 he was also a full professor at the Technical University in Breslau . From 1915 to 1918 he served in the army. From 1921 he was a professor in Frankfurt am Main .

When the Nazis came to power, he was released in 1935 and after the persecution of the Jews during the Reichskristallnacht he found refuge with Willy Hartner . In 1939 he fled Germany first to Copenhagen , then to Trondheim and finally to the USA. Due to the large number of emigrated scientists, he could not find a job there, and after short-term positions at Idaho Southern University (now Idaho State University ), the Illinois Institute of Technology and St. John's College in Annapolis (Maryland), he took a position at the artists' college Black Mountain College , where he was the only mathematician.

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The third Hilbert problem has its starting point in the fact that an elementary theory of the content of rectilinearly limited figures can be developed in two-dimensional Euclidean geometry according to Euclid (that is, without the use of boundary processes of calculus only by decomposition and composition from basic elements such as triangles), the Efforts by mathematicians to develop such in three dimensions, however, failed. In addition to the volume, Dehn found another invariant for polyhedra in three dimensions, which is retained in elementary decomposition and assembly processes ( Dehn invariant ). The angles of neighboring sides of the polyhedron and the edge lengths go into this invariant. He was then able to show that this was different for cubes and tetrahedra with the same content, which showed that they could not be converted into one another by elementary operations. The problem had been dealt with by the French mathematician Raoul Bricard before Dehn in 1896 and "almost" solved in a similar way (Dehn knew Bricard's work and quoted it).

Together with Poul Heegaard, Dehn wrote one of the first systematic reviews of topology (then called Analysis Situs ) in the Encyclopedia of Mathematical Sciences in 1907. Starting from topological questions, he also dealt with combinatorial group theory , where in 1911 he formulated the word problem for finitely generated groups : there there an algorithm to decide whether a word (product of generators) is equivalent to identity? In 1955 it was shown by Pyotr Sergeyevich Novikov that the problem is generally undecidable. In the same essay he also formulated other problems such as the isomorphism problem .

In 1910 a paper appeared in which Dehn gave a proof for a fundamental theorem of knot theory (that a knot is trivial if the fundamental group of the complement of the knot is cyclic), which later ( Hellmuth Kneser 1929) proved to be incomplete. The lemma of Dehn used in the proof was only proven in 1957 by Christos Papakyriakopoulos , with new methods that also revived the topology of three-dimensional manifolds. Some basic techniques in low-dimensional manifold topology are named after Dehn ( Dehn-Twist , Dehn-Surgery ).

In addition to his fundamental work on geometry and topology, he was also intensely interested in the history of mathematics, especially during his time in Frankfurt, where he led a related seminar with Carl Ludwig Siegel and others.

Namesake

• Stretching surgery
• Dehn invariant
• Stretch twist
• Dehn's lemma
• Dehn-Nielsen-Baer theorem
• Stretching function

Fonts

• Max Dehn: Papers on group theory and topology. Springer 1987. (Editor John Stillwell )

literature

• Wilhelm Magnus: Max Dehn. In: The Mathematical Intelligencer. Vol. 1, 1978/9, p. 132.
• Ruth Moufang, Wilhelm Magnus: Max Dehn in memory. In: Mathematical Annals. Vol. 127, 1954, pp. 215-227.
• R. Sher: Max Dehn and Black Mountain College. In: Mathematical Intelligencer. Vol. 16, 1994, p. 54.
• Wilhelm Süss:  Dehn, Max Wilhelm. In: New German Biography (NDB). Volume 3, Duncker & Humblot, Berlin 1957, ISBN 3-428-00184-2 , p. 565 f. ( Digitized version ).

Web links

• Literature by and about Max Dehn in the catalog of the German National Library
• Detailed information on Max Dehn (PDF file; 733 kB)
• History of Mathematics at the University of Münster, a. a. Biography of Dehn, pdf
• John J. O'Connor, Edmund F. Robertson :  Max Dehn. In: MacTutor History of Mathematics archive .
• Wolfgang Schwarz, Jürgen Wolfart, Gerhard Burde: Max Dehn and the Mathematical Seminar. (in Frankfurt), PDF file
• Max Dehn: his Life, Work, and Influence (Mathematisches Forschungsinstitut Oberwolfach Report No. 59/2016)

Individual evidence

1. ^ Wilhelm-Gymnasium Hamburg , 1881–1981, Höwer Verlag, Hamburg 1981, ISBN 3-922995-00-4 , p. 279.
2. ↑ Dehn: About the volume. In: Mathematical Annals. Volume 55, 1901, pp. 465-478.
3. ↑ Jeremy Gray : The Hilbert Challenge. Oxford University Press, 2000, pp. 98f.
4. ^ Dehn: About infinite discontinuous groups. In: Mathematical Annals. Volume 71, 1911, pp. 116-144.
5. ↑ Dehn: About the topology of three-dimensional space. ( Memento of the original from January 26, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. In: Mathematical Annals. Vol. 69, 1910, pp. 137-168. 
6. ^ Siegel: On the history of the Frankfurt Mathematics Seminar. In: Mathematical Intelligencer. Vol. 1, 1978/9, No. 4; Also Wolfgang Schwarz, Jürgen Wolfart: On the history of the mathematical seminar of the University of Frankfurt. 2002, ps file ( memento of the original dated June 11, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.