Max Dehn by David Peifer

An Artist among Mathematicians and a Mathematician among Artists

 

David Peifer

Introduction

Max Dehn was unique among the faculty at Black Mountain College. His great works are not paintings, music, or performance. His career and achievements are in theoretical mathematics. This article is meant to provide a brief history of Dehn’s life and career in mathematics, for those with an interest in Black Mountain College who may not have much mathematical background. When Dehn began at BMC, he was finishing an exemplary career as a mathematician. Why did Dehn end up teaching at BMC, an experimental liberal arts college with an emphasis on the arts? As a scientist, what did he bring to this unique experiment in liberal education? What did he have to add to the artistic and intellectual community? To begin to address these questions, it is necessary to first know Dehn’s history as a mathematician. As his student W. Magnus has said, “being a mathematician was an essential part of [Dehn's] personality”; he adds that, “it influenced also his very well founded and deep interests in the humanities, in art, and in nature.” [M] No one would study any of the painters of BMC without seeing their paintings, or composers without hearing their music. In this same way, to understand Dehn’s significance at BMC, it is important to examine his place in the history of mathematics.

During his life, Max Dehn was regarded as one of the great mathematicians of his time. He was the first to prove one of Hilbert’s famous problems. He co-authored the first survey article on topology and by doing so developed a framework and perspective of this topic that is still significant today. Dehn was skilled at finding insightful pictures and simple geometric representations of important mathematical ideas. His greatest mathematical works synthesized current knowledge and laid a foundation for future study.

Dehn began his career, at the turn of the century, in the mist of a grand project at the foundation of rational thought. Dehn’s advisor was leading a program to find the basic axioms on which all mathematics could be based. By the end of Dehn’s career, it would be shown that this program was bound for failure from the very beginning. There is no perfect set of axioms. To the dismay of the mathematical world, it was shown that in mathematics there are unsolvable problems. Dehn’s place in this story is significant.

Dehn’s insight and creativity has continued to influence the mathematical world. Shortly after he died, a problem he had posed over three decades earlier was answered in a way no one would have ever expected. Dehn’s famous Word Problem for Groups was the first “real” mathematics problem to be proven unsolvable. In the 1960′s Dehn’s techniques for studying the Word Problem were generalized and a new branch of mathematics began called Small Cancellation Theory. One of the most important results in Knot Theory was solved by Cameron Gordon, in 1989. Gordon uses fundamental ideas motivated by Dehn’s work on knots from 1914. In the 1980′s a leading geometer of our time, M. Gromov, published several papers laying out a new prospective on the study of infinite groups. Gromov’s work is clearly motivated by Dehn. In fact, Gromov proves that his new class of groups, called hyperbolic groups, are exactly the same as the set of groups that solve their word problem with a Dehn Algorithm. Dehn had used a similar idea in his study of 2-dimensional surfaces from 1910. The topologist, W. Thurston, now leads the mathematics community in a program to understanding three manifolds (three dimensional shapes sitting in four, or higher, dimensional space). His work uses a concept called Dehn surgery. Thurston has also done ground breaking work on infinite groups. His definitions of automatic groups is based on algorithmic structures that exemplify the algorithmic view of Group Theory Dehn first proposed almost a century earlier.

Since his death, several of Dehn’s mathematical works have been republished and translated. As recently as 1987, J. Stillwell translated into English several of Dehn’s important papers written between 1910 and 1914 [St]. Over the years, mathematical books and journals have published some very non-technical historical accounts of Dehn’s mathematical career, for example, [M], [Si], [Sh], [Da], and [Y]. The first of these is an account of Dehn’s career by one of his most successful students. The next is an account by a close friend and colleague of many years. The author of the last article talked with many of Dehn’s family and friends, including a student from BMC. I am in debt to these authors for much of the personal information about Dehn in this article. Dehn’s mathematics history articles from 1943-44 are also very accessible to the non-mathematically trained.

As Dehn was famous for his way of picturing abstract concepts, many of his great mathematical ideas can be intuitively understood through pictures. An effort has been made to present mathematical ideas, in a non-technical manner, accessible to a general audience. On first reading, skim through the mathematical details, particularly in Sections Two through Four. Section One provides some details of Dehn’s time at BMC. Section Two gives some background on the history of mathematics leading up to Dehn’s first works in geometry. Section Three is a brief account of the foundational work Dehn did in helping to create and focus the new mathematics discipline now known as topology. Section Four discusses Dehn’s work in algebra, in particular his work on infinite groups. This section includes a discussion of Dehn’s famous Word Problem and its historical significance. Section Five is devoted to Dehn’s time as Chair of Mathematics at the University of Frankfurt. The final section provides an account of Dehn and his wife’s flight from the Nazis in Germany, leading them eventually to the community at BMC.

Section 1: Time at Black Mountain College

Max Dehn came to Black Mountain College in March of 1944 to give two invited talks on the philosophy and history of mathematics. These talks must have been received well, because the next year Dehn was asked to join the faculty. At age 66, Dehn had behind him a long and illustrious career in mathematics. He had been the Chair of Mathematics at two German Universities. He had written definitive texts setting the foundations in three different branches of mathematics. He had solved some of the most well known mathematics problems of his time. His acceptance of a position at a small experimental college, with an emphasis on art, tucked back in the mountains of western North Carolina, has confused many mathematicians and writers. Some accounts of Dehn’s years in the United States simply dismiss his time at BMC as a mistake of the mathematics community that should have assured Dehn a position at a highly acclaimed research university. His eventual position at BMC is attributed to the poor economy at the time combined with Dehn’s late immigration compared with many other WWII refugee scientists looking for academic jobs. However, in all the accounts written by colleagues and students who actually knew him, there is agreement that Max Dehn found a home at Black Mountain College.

After negotiating the salary from $25 up to $40 dollars a month, Dehn accepted the position and started at Black Mountain College in January of 1945. The position included living arrangements on the campus for him and his wife, Toni. Dehn taught Mathematics, Philosophy, Greek, and Italian. In Mathematics his courses included History of Mathematics and Projective Geometry. He is remembered for his love of nature and the arts. His lectures frequently included tangents on philosophy, the arts, and their connections to mathematics. He was best when lecturing in a socratic style. He was fond of giving these lectures while hiking with his students through the woods. He is remembered as a caring and devoted teacher. He also seemed to get along with most of the other faculty. In many stories of conflicting views and egos from BMC, Dehn’s name comes up as a mediating voice in the discussion. Dehn and his wife were close friends with Buckminster Fuller.

Of the many BMC students who took courses from Dehn, three stand out more distinctly as Dehn’s students. Two have gone on to get PhD’s in Mathematics. Peter Nemenyi received a PhD in Mathematics at Princeton University, in 1963. His BMC exam was given by the prominent mathematician Emil Artin, from the Mathematics Department at Princeton. Peter’s social convictions led him to teach mathematics and statistics in underprivileged communities where the need was high, such as Tupelo, Mississippi, during the 60′s, and Sandinista, Nicaragua.

Another BMC student, Trueman MacHenry received a PhD in Mathematics from Adelphi University, in 1961. His BMC examiner was Ruth Moufang, one of Dehn’s former German PhD students. MacHenry was a well rounded student, taking classes in philosophy, writing, linguistics, French, German, and Russian. He also studied physical sciences with Natasha Goldovski and modern dance with Merce Cunningham and Katherine Litz. MacHenry’s mathematics classes with Dehn were one-on-one discussions, “sometimes in a room, sometimes while hiking.” Yandell, who spoke with MacHenry in 2000 about Dehn, reports “[MacHenry] said he never had another teacher who was as sensitive, deep, or vastly philosophic.” MacHenry followed a more traditional career in mathematics than Nemenyi, he became a mathematics professor at York University in Canada.

Dorothea Rockburne was also among Dehn’s students. She has gone on to a successful career as an artist. Her accomplishments include induction into the American Academy of Arts and Letters in 2001 and receiving the Academy Museum Artist’s Lifetime Achievement Award in 2009. Her work exemplifies the connections between art and science and is influenced by modern geometry and astronomy. She claims that Dehn was her greatest influence at BMC. Coming to BMC already having artistic training as a painter, she had more to learn from Dehn and his insights on the philosophy and history of science than from many of the art teachers. She credits Dehn for her fascination with mathematics and science.

Dehn retired from BMC and became an Emeritus faculty in the summer of 1952. He planned to remain in an advisory position and continue to live on the campus. He and his wife also had plans to visit his old university in Frankfurt for the first time after the war. Tragically however, that same summer Dehn died of a coronary embolism. Several authors have suggested a connection between his death and the stress he felt over mistakes that were made when the college arranged to sell some timber to raise needed money. Dehn now rests in peace, under a thicket of rhododendrons, on the old BMC campus.

Section 2: Beginnings in Geometry

Max Dehn was born on November 13, 1878, in Hamburg, Germany, one of eight siblings in a secularized Jewish family. His father, Maximilian Moses Dehn, was a physician. After graduating from the gymnasium in Hamburg, Max went on to the university. He began at Freiburg and then finished at Gottingen, where he studied with one of the great geometers of the time, David Hilbert.

Geometry is the study of two-dimensional shapes (points, lines, triangles…) and three dimensional volumes. With the high level of abstraction that has prevailed in the last century, mathematicians now study many dimensions far higher then three. Its theory relies heavily on the idea of a distance between points in space, called a distance function. Geometry has been a corner stone of mathematics since the time of Euclid (300BCE). In Euclid’s classic text The Elements, the first axiomatic approach to mathematics is presented. In Book I of this text, Euclid begins with five simple axioms (statements taken as true) and using basic logic is able to prove many geometric theorems. For example Proposition 47 is the famous Pythagorean Theorem, which, in modern terms, says that the sides of a right triangle most satisfy x^2+y^2=z^2.

Mathematics built on Euclid’s five axioms is called Euclidean Geometry. Euclid’s first four axioms were very simple and constructive in nature. For example, one axiom states simply that given two distinct points, a straight line can be draw from one point to the other. For centuries mathematicians tried to prove Euclid’s fifth, more complicated, axiom. Finally in 1830, Nikolai Lobachevsky discovered a new geometry, now called Hyperbolic Geometry. Around the same time, Carl Gauss and Janos Bolyai also independently made the same discovery. These mathematicians had found that the fifth axiom was independent of the first four. That is, they showed that the fifth axiom could not be proven from the first four, and that there were other non-Euclidean geometries that exist for which the first four axioms are true, but a different fifth axiom holds. The discovery of Hyperbolic Geometry shook the mathematical world. Euclidean Geometry had set the foundation of mathematics for 2000 years. If for centuries mathematicians had failed to notice a whole other geometry, what other flaws might be buried in the foundations of mathematics?

This discovery sparked an overhaul of the foundations of mathematics. Much of the work in the 19th century was done to shore up the most basic logical foundations of mathematics. Max Dehn’s advisor David Hilbert, is famous for his revised set of axioms for Euclidean Geometry. Hilbert’s extended set of axioms (15 in all) give a complete list of statements that must be assumed, as axioms, for Euclidean Geometry. Dehn’s early work is in the spirit of Hilbert’s work in geometry. In his PhD dissertation in 1900, Dehn showed that the Archimedean Postulate (an axiom that involves the idea of infinite limits) is essential to prove that the sum of the angles of a triangle does not exceed 180 degrees. Thus Dehn had shown that any proof of this simple result about triangles required the use of one of the deepest most sophisticated axioms, the Archemedean Postulate. One would have hoped that the idea of limits would not have been required.

Because of his renown, David Hilbert was asked to give the opening address at the International Congress of Mathematicians, held in 1900 in Paris. In this famous address, Hilbert presented 23 problems that he felt would shape the next 100 years of mathematics. (Only eight were actually presented at talk. The others appeared in the subsequent paper.) To this day, anyone who solves one of these problems is guaranteed a prominent position in the history of mathematics. In 1901, Dehn was the first to solve one of Hilbert’s problems. He solved the third problem. He showed that the Archimedean Postulate is also needed to prove that tetrahedra (a pyramid shape solid with four vertices) of equal base and height have equal volume. That is, he showed that the idea of infinite limits is needed to find the volume of a tetrahedron. The two dimensional analog of this problem, concerning the area of triangles, does not need this limit concept and was completely understood 2000 years earlier in Euclid’s time. For this work, Dehn was awarded his habilitation at the University of Munster. The habilitation is a degree beyond a PhD required in the German university system to become a faculty member.

During the next 35 years, Dehn had a very successful career in the German Mathematics community. From 1901 to 1911 Dehn was a Privatdozent (assistant professor) at the University of Munster. From 1911 to 1913 he was Extraordinarius (professor) at the University of Kiel. From 1913 to 1921 he was Ordinarius (professor and chair) of Mathematics at the University of Breslau. Finally, from 1921 to 1935 Dehn was Ordinarius (professor and chair) at the University of Frankfurt.

Section 3: Work in Topology

The first decades of the twentieth century were an exciting time in mathematics. Logicians worked to solidify every small gap in the basic foundations of mathematics and hoped to build a new edifice for mathematics based on Set Theory. In this vein, Bertram Russell and Alfred Whitehead began a program to find the definitive list of axioms that could be used to prove all of mathematics. Their work in the area appears in the three volume text, Principia Mathematica, from 1910. During the entire nineteenth century, analysis, the theory that lies beneath calculus, was rigorously investigated. By the beginning of the twentieth century, it was found that many ideas from analysis could be understood completely without need for the concept of a distance function. This was the birth of topology. Topology can be thought of as rubber sheet geometry. In this discipline, shapes are investigated. One is allowed to stretch and deform the shape as long as the shape is not torn or cut. The distance between points is not important, as it is in geometry. Two shapes that can be deformed into each other are considered equivalent, or the same. This is why, to a Topologist, the surface of a coffee cup is exactly the same as the surface of a donut. It is the hole in the handle that gives the cup its distinctive shape; the indentation that holds the coffee is inconsequential. Mathematicians, such as August Ferdinand Mobius, Felix Klein, and Henry Poincari, began developing concepts to help study shapes in this more modern view of topology.

An important problem at the beginning of topology was to classify all possible surfaces (2-dimensional shapes without boundaries). A similar problem had been done in geometry in the preceding century when it was shown that the 2-dimensional geometries (of constant curvature) are exactly Euclidean, Spherical, and Hyperbolic. In order to address the topological question, in 1895, Henri Poincare introduced the concept of the fundamental group of a surface. He showed that this group is a topological property. That is, the fundamental group of a surface does not change by stretching and bending the surface as long as the surface is not cut or torn.

The concept of groups was first formally defined by Arthur Cayley in a paper from 1854. A group, in mathematics, is a simple algebraic structure. It consists of a set of objects and an operation that takes two objects and combines them to get another object in the set. A basic example is the set of integers Z={ …,-2,-1,0,1,2,…} with the operation of addition. To qualify as a group, the set and its operation must satisfy three basic axioms. There must be an identity element. This is an element in the set that when combined, by the operation, with any other element gives as a result the second element back again. For example, in Z, with addition, 0 is the identity. Add 0 to any integer and the result is simply the same integer, that is, a+0=a. The second axiom requires that every element have an inverse. That is, for any element, there is another element that when combined with the first, by the operation, results in the identity. For example, the inverse of 3 is -3 since 3 + (-3)= 0. The last axiom is called that associative property and requires that (a+b)+c= a+(b+c). For the integers, this is a property that we learned in grade school arithmetic.

Poincari’s fundamental group of a surface took as elements the set of loops on the surface, starting and ending at a given base point. Two loops are considered equivalent if one can be deformed, on the surface, into the other without breaking or cutting the loop in the process. For example consider the torus, the surface of a donut shaped object, pictured in figure 1. First, note that the torus is only the outside skin of the donut, not the whole volume. Loops must stay on this outside surface. So there are two holes in the torus, the first is the obvious hole in the center. The second is the less obvious hole that makes up the core of the solid shape. In figure 1, the loop C may easily be deformed down to the point P without ever breaking the loop. So the trivial loop, consisting of only the base point, is equivalent to loop C. However, a loop that circles the hole in the center, A, can never be deformed down to the point. In fact, the loop that circles a meridian of the torus, B, can never be deformed to the loop A or the trivial loop P. So A, B, and C are in fact distinct non-equivalent loops. The operation on the set of loops is defined to be the resulting loop (or we should say the equivalence class of loops) obtained by following the first loop from the base point back to the base point again and then following the second loop back again to the base point.

From the beginning of topology around the turn of the century, topology and geometry were closely linked. Poincari, Klein, and others had noticed the connections. In particular, topologists were working to classify all 2-dimensional surfaces. They observed that the 2-dimensional topological surfaces (closed, bounded, and orientable surfaces) corresponded to specific tessellations of the hyperbolic plane. Poincari had given a nice way to picture the unbounded hyperbolic plane. In his work, Dehn was able to exploit this connection to provide the first rigorous proof classifying all the (closed, bounded, orientable) surfaces.

In 1907, Dehn and Poul Heegaard, published the first comprehensive article on topology (which at that time was called Analysis Situs). This article appeared in the first edition of the German Encyclopedia of Mathematical Science. Dehn’s contribution to this work is very visual and combinatorial. While this work is now outdated, it provided the basis and direction for research in this area that still continues today.

Of particular importance in topology is the study of knots. In mathematics, a knot is considered a closed curve in space. That is, a knot is a path (1-dimensional curve) in 3-dimensional space that begins and ends at the same point, and otherwise never intersects itself. Figure 2 is an example of a knot table, where all the knots with 5 or less crossings are presented. These pictures are called knot projections. The breaks in the paths are meant to indicate that the path crosses under itself. Every knot has many different projections. The problem of classifying all knots has been an important problem in mathematics since Dehn’s time. One of Dehn’s great works was to show that the right and left trefoil knots are in fact distinct. The two knots in figure 3 are the left and right trefoil knots. Dehn was able to prove his result using Poincari’s concept of a fundamental group. Consider a solid ball with the knot inside. Thicken up the path so that it is a cord with a small diameter. Now consider the ball with the knot missing, i.e. the ball now has a hollow wormhole twisting through it where the thickened knot had been. This is called the knot complement. The fundamental group of the knot complement is the set of (equivalence classes) of loops that can wrap around the missing knot, but may not pass though it. Dehn was able to analyze the fundamental group of the right and left knot complements to prove that the knots are topologically distinct. In 1989, a complete classification of knots was finally achieved by Cameron Gordon and John Luecke. Dehn’s original approach, of considering the fundamental group of the knot complement, played the central role in the work.

Section 4: Work in Algebra and the Word Problem

We have discussed Dehn’s work in geometry and in topology. It is now time to give a brief account of his work in algebra. Dehn’s work in algebra is in the area of infinite groups. The groups that arise in topology, as fundamental groups of a surface, are in general infinite. That is, the set of group elements is infinite. Up until the early 1900′s most group theory was concerned with finite groups. One of Dehn’s most important contributions to the study of infinite groups was to make serious use of the Cayley graph of a group. This is a pictorial representation of the group that had been proposed earlier by Cayley who had used them when studying finite groups. Figure 4 is the Cayley graph of the fundamental group of a two-holed torus. Each vertex corresponds to a group element and edges represent the group operations.

Groups in topology arise as a set of generators and a set of relations on the generators. The generators are group elements such that every group element can be obtained by performing the operations on combinations of the generators. Relations are rules that show properties of the operation. For example, in the fundamental group of a torus, if A and B, from figure 1, are taken as generators, then it can be shown that AB=BA. That is, the path going around A and then B can be deformed into the path the goes around B first and then A. The equation AB=BA is a relation for the torus group.

In his paper of 1912, Dehn poses three basic questions about groups given by generators and relations (as arises in topology). The first of these is called the word problem for groups. It asks if given a list of generators, called a word, is the product of this list, as an element in the group, actually the identity element. In the topological interpretation, this is to ask if a loop, defined as a product of generating loops, is actually equivalent to the trivial loop, i.e. can the loop be deformed back to the single base point? This is now known as the Word Problem for Groups.

Dehn was able to solve the word problem for a special class of groups called the surface groups. He did this using combinatorial (or geometric) properties of the Cayley graph. He was able to make a connection between the algebra of products of group generators with the geometry of paths through the Cayley graph. For the surface groups, he determined an algorithm for shortening a path in the Cayley graph. This corresponded to shortening the word, however continuing to represent the same group element. This algorithm will terminate in the empty word (single point path) if the original word is equivalent to the identity in the group. An algorithm of this type in infinite group theory is now known as a Dehn Algorithm.

Dehn’s work in this area has lead to a whole branch of mathematics called Combinatorial Group Theory. In particular, the work by Grindlinger, Lyndon, and Schupp, in the 1960′s, was motivated by Dehn’s solution to the word problem for the surface groups. These same ideas also play a motivational role in the theory of Hyperbolic Groups presented by M. Gromov in the 1980′s.

As I have mentioned before, in the early 1900′s mathematicians felt they were near to finding the ultimate set of axioms on which all of mathematics could be built. From this set of axioms, together with basic logic, all true statements about mathematics would be provable. The Hilbert Program was a plan initiated by Dehn’s advisor, David Hilbert, to find such a set of axioms and to eventually base all mathematics on the simple axioms of arithmatic. The fact that some mathematical problems would turn out to be unsolvable had not occurred to many. In fact, even a clear definition of what unsolvable means was not made precise until the work of Alan Turing, Alonzo Church, and Emil Post, in the 1920′s and 30′s.

In an earth shaking paper from 1931, Kurt Godel proved that for any finite set of axioms, which include the axioms of basic arithmatic, there are true mathematical statements that can not be proven true. No perfect set of axioms will ever exist. There will always be true statements that can not be justified by the axioms and logic. This is known as Godel’s Incompleteness Theorem. It shocked the mathematical world. At the time, most mathematicians felt that Godel’s ideas were an exotic philosophical paradox and that no “real” math problems would be unsolvable.

In 1952 and 1955, shortly after Dehn’s death, it was shown, independently by Novikov and Boone, that the Word Problem for groups is unsolvable. They had shown that there is a finitely presented group for which no algorithm could solve the Word Problem. From that time on, much of the work on infinite groups has involved ideas related to unsolvability and logic. What is significant here is that Dehn had realized the algorithmic questions and nature of infinite groups long before the logical rigor existed to address these questions.

Section 5: Frankfurt and the History Seminar

In his second year at Frankfurt, Dehn began a seminar on the History of Mathematics, which he continued to lead until his forced dismissal in 1935. There is a wonderful article about this seminar at Frankfurt written by Dehn’s close friend and colleague, Carl Ludwig Siegel. An English translation appears in the Mathematical Intelligencer, [S]. The article contains personal accounts of Dehn’s life during these times. The seminar usually consisted of department faculty and about 15 students. Siegel reports that the requirement of reading original works, in Greek, Dutch, and Italian, had a tendency to self select bright students. Over the 13-year period, they read works by Euclid, Archimedes, Descartes, Kepler, Fermat, and many others. The faculty in the seminar were not only good mathematicians, but also good friends. Dehn had always been an amateur naturalist, with a love for plants and the outdoors. Besides leading the history seminar, Dehn also frequently led hikes for students and faculty into the surrounding wilderness. The faculty in Frankfurt were dedicated mathematicians who shared their love for mathematics as well as their passion for life with their students. This lead to a productive healthy intellectual community that Siegel says he has never seen at any other institution.

While at Frankfurt, Dehn had seven PhD students. I will mention only two of these here. Wilhelm Magnus received his PhD is 1931 and later taught at Frankfurt until 1938. One of his early great works was the solution to the word problem for one relator groups. Because he refused to join the Nazi party, Magnus was not allowed to hold an academic position until after the war. In 1948 he immigrated to the United States. He taught at the Courant Institute of Mathematical Sciences, in New York University until 1973, and then at the Polytechnic Institute of New York, before retiring in 1978. Magnus’ work is primarily in the study of infinite groups and much of his work relates to concepts derived from techniques initially investigated by Dehn.

Another of Dehn’s PhD students was Ruth Moufang, who also completed her PhD in 1931. Her dissertation was on Projective Geometry. She went on to do ground breaking work in geometry. Because she was a woman she was not allowed to teach by the Nazis. But after the war, in 1947, she was hired as a mathematics professor at the University of Frankfurt. She was the first woman in Germany to become a full professor in Mathematics.

Section 6: From Germany to America

After the Nazis took over the government in 1933, Dehn continued to teach. Many Jews had already been removed from academic positions. However, because of his service to Germany in the First World War, Dehn was allowed to continue in his position. In 1935, all Jews were force to leave the university by the Nazis. Dehn and his wife remained in Germany. In 1936, he sent his children to study in the United States and England. On the morning after Kristallnacht, November 11, 1938, Dehn and his wife, along with tens of thousands of other Jews, were arrested. After a day in holding, the Dehns were sent home because there was no more room to hold prisoners. The next day they went into hiding.

First they stayed with one of Dehn’s former colleagues, Willi Hartner. Here Dehn celebrated his 60th birthday. Hartner later reflected on Dehn’s calmness that day, stating that “the conversations centered not on the events of the day, but on the relationship between mathematics and art, on problems of archaeology, and finally on the concept of the humanity of Confucius.” The Dehns were then helped by the family of his former student Wilhelm Magnus and by Dehn’s sister, who, because of her old age, had not yet been harassed by the Nazis. With the help of Carl Siegel, the Dehns escaped to Denmark and then to Norway. Dehn had been a frequent visitor to Norway and was fluent in the language. In Trondheim, Norway, Dehn was able to secure a visiting position at the Norwegian Institute of Technology (Technische Hochschule). However, after the German invasion of Norway, in October of 1940, Dehn and his wife left for America. They traveled by way of the Trans-Siberian railway and then through Japan and across the Pacific. There is an article comparing Dehn’s and Godel’s experiences leaving Germany in the Notices of the American Mathematical Society, from October of 2002, [Da]. Dehn and his wife Toni arrived in San Francisco on March 4, 1940 and took the train across the country to Princeton. By the time they reached America, they had lost all their savings and possessions.

Dehn was unable to secure a permanent research position. Siegel suggests that despite his strong reputation, the financial crisis at the time made it hard for prestigious universities to offer him a deserving salary. So rather than offer him an ill-paying position, they found it easier to simply ignore him. For one and a half years, from 1941 to 1942, Dehn held a visiting position at the University of Idaho, in Pocatello (now known as Idaho Southern University). This position had been arranged for him by friends and was necessary in order for him to legally immigrate to the United States. Dehn had to immediately begin searching for another position. He found another visiting position at the Illinois Institute of Technology, in Chicago, Illinois. According to Siegel, Dehn never felt comfortable in the city and was unhappy. Dehn then moved on to become a Tutor at the prestigious Liberal Arts College, St John’s College, in Annapolis, Maryland. This college has a unique curriculum, based on the Great Books Program from the University of Chicago. Students read texts from world literature. Siegel comments that because it was during the war, the students attending the college were very young and not prepared enough for such a demanding endeavor of reading the original texts of Homer, Dante, Descartes, and Goethe. Dehn apparently voiced his concerns about the curriculum to the college administration, which did not go over well.

In March of 1944, Dehn came to BMC and gave the two talks mentioned at the beginning of this paper. As a hiker and amateur naturalist, he must have fallen in love with the mountains at first sight. As a true intellect of his time, he must have hungered for the contact with the artists and thinkers at the college. As a great teacher, he must have realized the chance to teach in this unique free and energetic learning environment. In many ways, Dehn was a perfect fit at BMC.

 

 

 

 

 

 

Footnotes:

Mary Emma Harris generously offered the following information in regards to the Dehns at BMC: “I question whether the Dehns were close friends of Fuller.  They were there together but I have no reason to believe they were close friends although at one time I might have assumed they were.  Some students say Dehn was very skeptical of Fuller’s theories.  Somewhere I have a note that he Dehn’s had bought a house in Black Mountain and Planned to move there.  It surprised me since they did not, to my knowledge, have any money.”

References:

[Da] by John W. Dawson Jr., Max Dehn, Kurt Godel, and the Trans-Siberian Escape Route, in the Notices of the AMS, no. 9, pages 1068-1075, 2002

[De] by Dehn, M., Translated by John Stillwell, Papers on Group Theory and Topology, Springer-Verlag, 1987

[Du] by Martin Duberman, Black Mountain and Exploration in Community, 1972

[G] by Gromov, M., Hyperbolic groups, in Essays in Group Theory, ed. by S.M. Gersten, pages 75-264, Springer-Verlag, 1987

[GM] by Grossman, I. and Magnus W., Groups and Their Graphs, in the New Mathematical Library no. 14, The Mathematical Association of America, 1964

[H] by Mary Emma Harris, The Arts at Black Mountain College, MIT Press, 1987

[LS] by Lyndon, R. and Schupp, P.E., Combinatorial Group Theory, Springer-Verlag, 1977

[M] by Wilhelm Magnus, Max Dehn, in The Mathematical Intelligencer, vol. 1 no. 3, pages 132-143, 1978

[MKS] by Magnus, W., Karrass, A. and Solitar D., Combinatorial Group Theory, Dover, 1976

[Sh] by R. B. Sher, Max Dehn and Black Mountain College, in The Mathematical Intelligencer, vol. 16 no. 1, pages 54-55, 1994

[Si] by Carl Ludwig Siegel, On the History of the Frankfurt Mathematics Seminar, The Mathematical Intelligencer, vol. 1 no. 4, pages 223-230, 1979

[St] by Stillwell, J., Classical Topology and Combinatorial Group Theory, Springer-Verlag, 199

[Y] by Yandell, B. H., The Honors Class: Hilbert’s Problems and Their Solvers, A K Peters, 2002

 

A peer-reviewed publication of The Black Mountain College Museum + Arts Center