Having worked for a year in a bank in Albuquerque, Lyndon returned to Harvard, being awarded a Master's Degree in 1941. He taught at Georgia Institute of Technology in Atlanta during session 1941-42, then he returned to Harvard for the third time in 1942 and there taught navigation as part of the V-12 Program while he studied for his doctorate. The V-12 Navy College Training Program was set up in 1943 since the lowering of the draft age meant that there would otherwise have been a shortage of college trained officers. The men Lyndon taught went on to serve their country by manning ships, flying planes and commanding troops. On the research front Lyndon took some time working in different mathematical areas. He began working on mathematical logic, moved to a study of relational algebras, then finally undertook research in homological algebra. He was awarded a Ph.D. in 1946 for a thesis on homological algebra, the work being an outstanding early step in the study of spectral sequences. His supervisor was Saunders Mac Lane and his thesis was entitled The Cohomology Theory of Group Extensions. Mac Lane explained that the thesis addressed :-
... the problem of computing the cohomology groups of a group extension in terms of the cohomology groups of the factors of that extension. It turned out that in many cases one could not completely compute the whole cohomology group. Now we know that the difficulty lies in the additional invariants presented by the spectral sequence of that group extension.Interestingly Lyndon's first publication was not in any of the topics that we have mentioned above but rather was The Zuse computer (1947). In the paper he described the Z4, Zuse's relay-type digital computer which was discovered by advancing British and American troops,The nearly completed computer had been hidden by Konrad Zuse in the cellar of a house in the small village of Hinterstein in Bavaria. Lyndon's second paper The Cohomology Theory of Group Extensions was based on his doctoral thesis and appeared in print in 1948. In the following year the paper New proof for a theorem of Eilenberg and Mac Lane appeared, then, in 1950, the paper The representation of relational algebras which resulted from his early interest in that topic.
After attending a course by Alfred Tarski, Lyndon and Tarski became good friends and Lyndon was later to work on model theory as a result of attending these lectures. For two years after completing is doctorate, Lyndon worked for the Office of Naval Research in London. Returning to the United States in 1948 he decided that to make further progress in cohomology theory he needed to learn more about current work in topology and clearly Princeton was the leading institution for research in that area. Accepting a position as an Instructor in Mathematics at Princeton, he attended a course on knot theory by Ralph Fox and from this his interest was aroused in combinatorial group theory. Kurt Reidemeister was at Princeton for a year in 1948 and again this was a major influence on Lyndon to work on group presentations. Lyndon's first work which came out of these discussions with Reidemeister was published in 1950. In it Lyndon investigated one-relator groups. In particular he computed their cohomology groups.
In 1953 Lyndon left Princeton, where he had been promoted to assistant professor, and took up an assistant professorship at the University of Michigan where he remained throughout his career except for a number of posts as visiting professor at Berkeley, Queen Mary College, London, Montpellier, France and Picardie, France.
Kenneth Appel, writes in :-
Lyndon produces elegant mathematics and thinks in terms of broad and deep ideas.... I once asked him whether there was a common thread to the diverse work in so many different fields of mathematics, he replied that he felt the problems on which he had worked had all been combinatorial in nature... one would certainly have to put him in the very first rank of those who have used combinatorial techniques in the last forty years.Lyndon made numerous major contributions to combinatorial group theory. These include the development of 'small cancellation theory', work on Fuchsian groups and the Riemann-Hurwitz formula, his introduction of 'aspherical' presentations of groups and his work on length functions in free products of groups.
Lyndon was the coauthor of one of the most important works on combinatorial group theory. Together with Paul Schupp, he wrote Combinatorial group theory (1976). I [EFR] remember how eagerly the book was awaited by those interested in research in this area, and the excitement of seeing the book when it first appeared and was passed round a lecture theatre at a conference I was attending. Combinatorial group theory was not Lyndon's first book, however, for he had published Notes on logic ten years earlier in 1966. Andrzej Mostowski described the book, beginning his review:-
The book contains a concise and clear exposition of basic notions of mathematical logic. More specifically, the author deals with the first-order logic with symbols for relations and functions.After a detailed description of the contents, Mostowski continued:-
As should be clear from this description, the book is intended as a text to enable a mathematically well-trained reader to acquaint himself quickly with the basic results of logic. Owing to the clear plan, concise style and well-chosen exercises, the book meets its objectives excellently. In the reviewer's opinion the book has set a new standard for texts in mathematical logic which will not be easy to supersede.Lyndon's last book was Groups and geometry (1985). Kenneth S Brown writes:-
This book is a very readable introduction to group theory, geometry, and the connections between them. It is aimed at advanced undergraduates and beginning graduate students. The emphasis is on the classical two-dimensional incidence geometries and their associated groups. The geometries studied include Euclidean geometry, affine geometry, projective geometry, inversive geometry, and hyperbolic geometry. In addition, there are three optional chapters, treating plane crystallographic groups, tessellations in dimensions greater than 2, and Fuchsian groups.At Michigan, Lyndon supervised many Ph.D. students. The titles of their theses indicate the wide range of Lyndon's interests. They include: Groups rings and dimension subgroups; Two investigations on the borderline of logic and algebra; Decision problems of finite automata design and related arithmetic; On Dehn's algorithm and the conjugacy problem; Projectivities of free products; Continuous model theory and set theory; Real length functions in groups; Automorphisms of the fundamental group of an orientable 2-manifold; Some algorithmic problems for semigroups; and Groups acting on trees. Kenneth Appel, who was Lyndon's second Ph.D. student, writes :-
I fondly remember my experiences working as a thesis student in the years 1957-59. My conversations with Lyndon often took place in the newly established common room, which was supplied with a large coffee percolator. As a "morning person," I would want to discuss my work with Lyndon as soon as he arrived at Angell Hall. It soon became clear to me that no mathematical discussion was possible until he had consumed at least two cups of coffee, so during my last two years of graduate work I was the unofficial coffee preparer for the common room.
Article by: J J O'Connor and E F Robertson
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