Max Dehn

Max Dehn. He was a German-American mathematician who made foundational contributions to geometry, topology, and group theory. A student of the renowned David Hilbert, his work on the foundations of geometry and the topology of manifolds was highly influential. His career was dramatically disrupted by the rise of the Nazi Party, leading him to emigrate and eventually settle in the United States.

Biography

Born in Hamburg to a Jewish family, he began his university studies in Freiburg before moving to the University of Göttingen, then a world center for mathematics under Felix Klein and David Hilbert. He completed his Habilitation in 1911 and held positions at the University of Kiel and later the University of Frankfurt. With the enactment of the Nuremberg Laws, he was forced from his professorship in 1935. After perilous journeys through Norway and the Soviet Union, and a period of internment in a Finnish camp, he escaped via Siberia and Japan to the United States in 1940. He held temporary positions at several institutions, including the University of Idaho and Illinois Institute of Technology, before spending his final years at Black Mountain College in North Carolina.

Mathematical work

His early work solved the third of Hilbert's problems, proving that regular tetrahedra are not equidecomposable with cubes, introducing the powerful Dehn invariant. In topology, he pioneered the study of knot theory and 3-manifolds, developing fundamental tools like Dehn surgery and the Dehn twist. His work in combinatorial group theory was equally profound, where he formulated the critical word problem for groups and developed Dehn's algorithm for solving it in hyperbolic groups. He also made significant contributions to the study of infinite groups and the fundamental group of surfaces.

Dehn's problems

In 1911, he posed three fundamental decision problems in combinatorial group theory, now collectively known as Dehn's problems. These are the word problem, the conjugacy problem, and the isomorphism problem for finitely presented groups. The insolvability of the general word problem was later proven by Pyotr Novikov and independently by William Boone. Research into these problems and special classes of groups where they are solvable, such as automatic groups, has driven major areas of research in algebraic topology and theoretical computer science.

Influence and legacy

His ideas permeate modern low-dimensional topology and geometric group theory. Concepts like Dehn filling are central to the geometrization of 3-manifolds. The existence of Dehn functions measuring the complexity of the word problem is a key invariant in group theory. His life story is a poignant chapter in the history of mathematics during World War II, illustrating the intellectual diaspora caused by the Third Reich. His students, including Wilhelm Magnus and Ruth Moufang, carried forward his research traditions in Germany and beyond.

Selected publications

* "Über den Rauminhalt" (1900) in Mathematische Annalen, solving Hilbert's third problem. * "Über unendliche diskontinuierliche Gruppen" (1911) in Mathematische Annalen, introducing the word problem. * *Papers on Group Theory and Topology* (1987), an English translation of his major works edited by John Stillwell. * His collaboration with Poul Heegaard on analysis situs for the Encyklopädie der mathematischen Wissenschaften was a seminal early survey of topology.

Category:German mathematicians Category:Topologists Category:1878 births Category:1952 deaths