Max Dehn solved Hilbert's third problem in 1900, marking the first time one of the 23 Hilbert Problems reached a definitive conclusion. This early breakthrough established his influence in geometry, topology, and combinatorial group theory. Across a career spanning academic institutions in Germany, Norway, and the United States, he fundamentally reshaped the study of geometric group theory and three-dimensional topology.
Academic Foundations and Early Innovations
Born in Hamburg in 1878, Dehn pursued his education at the University of Göttingen, where he studied under David Hilbert. His habilitation work introduced the Dehn invariant, a crucial tool for analyzing polyhedral volume that resolved Hilbert's third problem. During his tenure at the University of Münster from 1900 to 1911, he proved the Jordan curve theorem for polygons and expanded his research into axiomatic geometry by constructing non-Archimedean Dehn planes.
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In 1907, Dehn co-authored the first foundation text on combinatorial topology with Poul Heegaard. His work in three-dimensional topology introduced the technique now known as Dehn surgery, which he utilized to construct homology spheres. In 1911, he formulated the word problem for groups, often called the Dehn problem. Subsequent contributions included the development of Dehn's algorithm for conjugacy problems and the estimation of relation areas through the Dehn function.
Teaching and Exile
Dehn held teaching positions at Kiel University and the University of Wrocław before assuming a professorship at Goethe University Frankfurt in 1922. During his time at Frankfurt, he led seminars on the history of mathematics that attracted prominent scholars such as Carl Ludwig Siegel and André Weil. After being forced to retire in 1935, he remained in Germany until 1939, when he escaped to Norway to teach at the Norwegian Institute of Technology.
Final Years at Black Mountain College
Dehn relocated to the United States in 1940, eventually joining the faculty of Black Mountain College in 1945. There, he taught diverse subjects including philosophy, Greek, and Italian. In his geometry courses for artists, he utilized the surrounding forested landscape for instruction. He remained an active advisor as Professor Emeritus until his death from an embolism in 1952. He is buried on the college campus.
Fast facts
- Born: 1878, Hamburg
- Died: 1952, Black Mountain
- Educated at: University of Göttingen
- Notable work: Dehn's lemma
- Citizenship: Germany, United States
- Final employer: Black Mountain College
- Married: Antonie Landau, 1912
Questions readers ask
What is the Dehn invariant?
It is a mathematical tool introduced by Dehn in 1900 to resolve Hilbert's third problem regarding the equality of polyhedral volumes.
Where did Max Dehn teach in the United States?
He held positions at the Illinois Institute of Technology, St. John's College, and Black Mountain College.
Achievements
- Notable work: Dehn invariant
- Notable work: Dehn twist
- Notable work: Dehn's lemma
- Notable work: Dehn surgery
- Affiliated with Goethe University Frankfurt, University of Münster and Illinois Institute of Technology
- Educated at University of Göttingen
- Worked as mathematician, historian of mathematics and topologist
