Max Dehn
Encyclopedia
Max Dehn was a German American
mathematician
and a student of David Hilbert
. He is most famous for his work in geometry
, topology
and geometric group theory
. Dehn's students include Ott-Heinrich Keller
, Ruth Moufang
, Wilhelm Magnus
and the artist Dorothea Rockburne
.
, Imperial Germany.
He studied the foundations of geometry
with Hilbert
at Göttingen
in 1899, and obtained a proof of the Jordan curve theorem
for polygon
s. In 1900 he wrote his dissertation on the role of the Legendre angle sum theorem in axiomatic geometry.
From 1900 to 1911 he was an employee and researcher at the University of Münster
. In his habilitation
at the University of Münster
in 1900 he resolved Hilbert's third problem
, by introducing what was afterwards called Dehn invariant.
Dehn's interests later turned to topology
and combinatorial group theory
. In 1907 he wrote with Poul Heegaard
the first book on the foundations of combinatorial topology
, then known as analysis situs. Also in 1907, he described the construction of a new homology sphere
. In 1908 he believed that he had found a proof of the Poincaré conjecture
, but Tietze
found an error.
In 1910 Dehn published a paper on three dimensional topology in which he introduced Dehn surgery
and used it to construct homology spheres. He also stated Dehn's lemma
, but an error was found in his proof by Hellmuth Kneser
in 1929. The result was proved in 1957 by Christos Papakyriakopoulos
. The word problem for groups
, also called the Dehn problem, was posed by him in 1911.
In 1912 Dehn invented what is now known as Dehn's algorithm and used it in his work on the word and conjugacy problems for groups
. The notion of a Dehn function
in geometric group theory
, which estimates the area of a relation in a finitely presented group in terms of the length of that relation, is also named after him. In 1914 he proved that the left and right trefoil knot
s are not equivalent. In the early 1920s Dehn introduced the result that would come to be known as the Dehn-Nielsen theorem; its proof would be published in 1927 by Jakob Nielsen
.
In 1922 Dehn succeeded Ludwig Bieberbach
at Frankfurt, where he stayed until he was forced to retire in 1935. He stayed in Germany until January 1939, when he fled to Copenhagen, and then to Trondheim, Norway, where he took a position at the Norwegian Institute of Technology
. In October 1940 he left Norway for America by way of Siberia and Japan (the Atlantic crossing was considered too dangerous).
In America, Dehn obtained a position at Idaho Southern University (now Idaho State University
). In 1942 he took a job at the Illinois Institute of Technology
, and in 1943 he moved to St. John's College
in Annapolis, Maryland
. Finally in 1945, he moved to Black Mountain College
, where he was the only mathematician. He and his wife regularly had long breakfasts with Buckminster Fuller
and his wife.
He died in Black Mountain
, North Carolina
in 1952.
German American
German Americans are citizens of the United States of German ancestry and comprise about 51 million people, or 17% of the U.S. population, the country's largest self-reported ancestral group...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and a student of David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
. He is most famous for his work in geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
. Dehn's students include Ott-Heinrich Keller
Ott-Heinrich Keller
Eduard Ott-Heinrich Keller was a German mathematician who worked in the fields of geometry, topology and algebraic geometry. He formulated the celebrated problem which is now called the Jacobian conjecture in 1939.He was born in Frankfurt–am-Main, and studied at the universities of Frankfurt,...
, Ruth Moufang
Ruth Moufang
Ruth Moufang was a German mathematician.Born to a German chemist Dr. Eduard Moufang and Else Fecht Moufang, she studied mathematics at the University of Frankfurt. In 1931 she received her Ph.D. on projective geometry under the direction of Max Dehn, and in 1932 spent a fellowship year in Rome...
, Wilhelm Magnus
Wilhelm Magnus
Wilhelm Magnus was a mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations....
and the artist Dorothea Rockburne
Dorothea Rockburne
Dorothea Rockburne is an abstract painter drawing inspiration primarily from her deep interest in mathematics and astronomy. In 1950 she moved to the United States to attend Black Mountain College, where she studied with mathematician Max Dehn, a lifelong influence on her work...
.
Biography
Dehn was born in HamburgHamburg
-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...
, Imperial Germany.
He studied the foundations of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
with Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
at Göttingen
Georg-August University of Göttingen
The University of Göttingen , known informally as Georgia Augusta, is a university in the city of Göttingen, Germany.Founded in 1734 by King George II of Great Britain and the Elector of Hanover, it opened for classes in 1737. The University of Göttingen soon grew in size and popularity...
in 1899, and obtained a proof of the Jordan curve theorem
Jordan curve theorem
In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...
for polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
s. In 1900 he wrote his dissertation on the role of the Legendre angle sum theorem in axiomatic geometry.
From 1900 to 1911 he was an employee and researcher at the University of Münster
University of Münster
The University of Münster is a public university located in the city of Münster, North Rhine-Westphalia in Germany. The WWU is part of the Deutsche Forschungsgemeinschaft, a society of Germany's leading research universities...
. In his habilitation
Habilitation
Habilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...
at the University of Münster
University of Münster
The University of Münster is a public university located in the city of Münster, North Rhine-Westphalia in Germany. The WWU is part of the Deutsche Forschungsgemeinschaft, a society of Germany's leading research universities...
in 1900 he resolved Hilbert's third problem
Hilbert's third problem
The third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the...
, by introducing what was afterwards called Dehn invariant.
Dehn's interests later turned to topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and combinatorial group theory
Combinatorial group theory
In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations...
. In 1907 he wrote with Poul Heegaard
Poul Heegaard
Poul Heegaard was a Danish mathematician active in the field of topology. His 1898 thesis introduced a concept now called the Heegaard splitting of a 3-manifold. Heegaard's ideas allowed him to make a careful critique of work of Henri Poincaré...
the first book on the foundations of combinatorial topology
Combinatorial topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces were regarded as derived from combinatorial decompositions such as simplicial complexes...
, then known as analysis situs. Also in 1907, he described the construction of a new homology sphere
Homology sphere
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is,andTherefore X is a connected space, with one non-zero higher Betti number: bn...
. In 1908 he believed that he had found a proof of the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
, but Tietze
Heinrich Franz Friedrich Tietze
Heinrich Franz Friedrich Tietze was an Austrian mathematician, famous for the Tietze extension theorem. He also developed the Tietze transformations for group presentations, and was the first to pose the group isomorphism problem.He was born in Schleinz, Austria and died in Munich,...
found an error.
In 1910 Dehn published a paper on three dimensional topology in which he introduced Dehn surgery
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
and used it to construct homology spheres. He also stated Dehn's lemma
Dehn's lemma
In mathematics Dehn's lemmaasserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disc's interior, implies the existence of another piecewise-linear map of the disc which is an embedding and is identical to the original on the boundary of the...
, but an error was found in his proof by Hellmuth Kneser
Hellmuth Kneser
Hellmuth Kneser was a German mathematician, who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition for 3-manifolds...
in 1929. The result was proved in 1957 by Christos Papakyriakopoulos
Christos Papakyriakopoulos
Christos Dimitriou Papakyriakopoulos, commonly known as "Papa" , was a Greek mathematician specializing in geometric topology. He worked in isolation at Athens University being awarded a Ph.D on the recommendation of Carathéodory...
. The word problem for groups
Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...
, also called the Dehn problem, was posed by him in 1911.
In 1912 Dehn invented what is now known as Dehn's algorithm and used it in his work on the word and conjugacy problems for groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. The notion of a Dehn function
Dehn function
In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in that group in terms of the length of that relation...
in geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
, which estimates the area of a relation in a finitely presented group in terms of the length of that relation, is also named after him. In 1914 he proved that the left and right trefoil knot
Trefoil knot
In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...
s are not equivalent. In the early 1920s Dehn introduced the result that would come to be known as the Dehn-Nielsen theorem; its proof would be published in 1927 by Jakob Nielsen
Jakob Nielsen (mathematician)
Jakob Nielsen was a Danish mathematician known for his work on automorphisms of surfaces. He was born in the village Mjels on the island of Als in North Schleswig, in modern day Denmark. His mother died when he was 3, and in 1900 he went to live with his aunt and was enrolled in the Realgymnasium...
.
In 1922 Dehn succeeded Ludwig Bieberbach
Ludwig Bieberbach
Ludwig Georg Elias Moses Bieberbach was a German mathematician.-Biography:Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doctorate in 1910. His dissertation was titled On the theory of automorphic functions...
at Frankfurt, where he stayed until he was forced to retire in 1935. He stayed in Germany until January 1939, when he fled to Copenhagen, and then to Trondheim, Norway, where he took a position at the Norwegian Institute of Technology
Norwegian Institute of Technology
The Norwegian Institute of Technology, known by its Norwegian abbrevation NTH was a science institute in Trondheim, Norway. It was established in 1910, and existed as an independent technical university for 85 years, after which it was merged into the University of Trondheim as an independent...
. In October 1940 he left Norway for America by way of Siberia and Japan (the Atlantic crossing was considered too dangerous).
In America, Dehn obtained a position at Idaho Southern University (now Idaho State University
Idaho State University
Idaho State University is a public university located in Pocatello, Idaho. It has outreach programs in Coeur d'Alene, Idaho Falls, Boise, and Twin Falls....
). In 1942 he took a job at the Illinois Institute of Technology
Illinois Institute of Technology
Illinois Institute of Technology, commonly called Illinois Tech or IIT, is a private Ph.D.-granting university located in Chicago, Illinois, with programs in engineering, science, psychology, architecture, business, communications, industrial technology, information technology, design, and law...
, and in 1943 he moved to St. John's College
St. John's College, U.S.
St. John's College is a liberal arts college with two U.S. campuses: one in Annapolis, Maryland and one in Santa Fe, New Mexico. Founded in 1696 as a preparatory school, King William's School, the school received a collegiate charter in 1784, making it one of the oldest institutions of higher...
in Annapolis, Maryland
Annapolis, Maryland
Annapolis is the capital of the U.S. state of Maryland, as well as the county seat of Anne Arundel County. It had a population of 38,394 at the 2010 census and is situated on the Chesapeake Bay at the mouth of the Severn River, south of Baltimore and about east of Washington, D.C. Annapolis is...
. Finally in 1945, he moved to Black Mountain College
Black Mountain College
Black Mountain College, a school founded in 1933 in Black Mountain, North Carolina, was a new kind of college in the United States in which the study of art was seen to be central to a liberal arts education, and in which John Dewey's principles of education played a major role...
, where he was the only mathematician. He and his wife regularly had long breakfasts with Buckminster Fuller
Buckminster Fuller
Richard Buckminster “Bucky” Fuller was an American systems theorist, author, designer, inventor, futurist and second president of Mensa International, the high IQ society....
and his wife.
He died in Black Mountain
Black Mountain, North Carolina
Black Mountain is a town in Buncombe County, North Carolina, United States. The population was 7,511 at the 2000 census. It is part of the Asheville Metropolitan Statistical Area. The town is named for the Black Mountain range of the Blue Ridge range in the Southern Appalachians.-History:Black...
, North Carolina
North Carolina
North Carolina is a state located in the southeastern United States. The state borders South Carolina and Georgia to the south, Tennessee to the west and Virginia to the north. North Carolina contains 100 counties. Its capital is Raleigh, and its largest city is Charlotte...
in 1952.
See also
- Cauchy's theoremCauchy's theorem (geometry)Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states thatconvex polytopes in three dimensions with congruent corresponding faces must be congruent to each other...
- Dehn's lemmaDehn's lemmaIn mathematics Dehn's lemmaasserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disc's interior, implies the existence of another piecewise-linear map of the disc which is an embedding and is identical to the original on the boundary of the...
- Dehn planeDehn planeIn geometry, Dehn constructed two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel to a given one that pass through a given point, but where the sum of the angles of a triangle is at least π. A similar phenomenon occurs in...
- Dehn twistDehn twistIn geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface .-Definition:...
- Dehn-Sommerville equationsDehn-Sommerville equationsIn mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their general form was established by Duncan Sommerville in...
- Scissors congruence
External links
- Dehn's archive – at the University of Texas at AustinUniversity of Texas at AustinThe University of Texas at Austin is a state research university located in Austin, Texas, USA, and is the flagship institution of the The University of Texas System. Founded in 1883, its campus is located approximately from the Texas State Capitol in Austin...
- "Max Dehn, Kurt Godel, and the Trans-Siberian Escape Route", an article by John W. Dawson in the Notices of the AMSNotices of the American Mathematical SocietyNotices of the American Mathematical Society is a membership magazine of the American Mathematical Society, published monthly except for the combined June/July issue. It is the world's most widely read mathematics magazine, sent to the approximately 30,000 AMS members worldwide...
.