Max Dehn

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

Life

Dehn studied at the Albert -Ludwigs- University of Freiburg and the Georg- August -Universität Göttingen. After he received his doctorate in Göttingen with David Hilbert with the dissertation The Legendre sentences about the angle sum in a triangle in 1900. In 1901 he completed his habilitation at the then " Academic educational institution " in Münster ( Westphalia ) and was there at the Westfälische Wilhelms- University lecturer until 1911. In this habilitation thesis, he has served as the first one (the third ) of Hilbert's 23 mathematical problems, but his performance was not very transparent and complicated and has been simplified by Veniamin Kagan and Hugo Hadwiger and completed. From 1911 he was an associate professor at the Christian -Albrechts -University of Kiel and in addition from 1913 full professor at the Technical University of Wroclaw. From 1915 to 1918 he made ​​military service. From 1921 he was a professor in Frankfurt am Main.

Due to the rise of the Nazis, he was released in 1935 and he left Germany in 1939 and fled first to Copenhagen and then to Trondheim. He then fled to the United States. There he found due to the many émigré scientists no body and finally increased after short-term sites on the Idaho Southern University (now Idaho State University), the Illinois Institute of Technology, at St. John 's College in Annapolis (Maryland ), a point at Artist College Black Mountain College, where he was the only mathematician.

Work

The third Hilbert problem has its starting point the fact that, although an elementary theory of content straight limited figures in the two-dimensional Euclidean geometry can be developed by Euclid ( that is, without use of limit processes of calculus only by decomposition and composition of basic elements such as triangles ), the efforts of mathematicians were but such a development in three dimensions failed. Dehn found in three dimensions in addition to the volume of a more invariant in polyhedra, which is maintained at elementary decomposition and composition processes ( Dehn invariant ). In this invariant the angle of adjacent sides of the polyhedron and the edge lengths a go. He was then able to show that this was different for the same content cubes and tetrahedrons, which has been shown that they could not be converted by elementary operations into one another. The problem had been treated before Dehn in 1896 by the French mathematician R. Bricard and "almost " solved in a similar manner ( Dehn knew Bricards work and quoted them).

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

In 1910 appeared a work in which Dehn proof of a basic set of knot theory was ( that a node is trivial if the fundamental group of the complement of the node is cyclic), which proved later ( Hellmuth Kneser 1929) to be incomplete. The lemma used in the proof of Dehn was not until 1957 proved by Christos Papakyriakopoulos, with new methods, the newly winged and the topology of three-dimensional manifolds. Some basic techniques in the topology of three-dimensional manifolds by Dehn are named ( Dehn twist, Dehn Dehn Surgery or surgery).

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

Eponym

• Dehn surgery
• Dehn invariant
• Dehn twist
• Dehn's lemma

Writings

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