Roger Conant Lyndon (December 18, 1917 – June 8, 1988) was an American mathematician, for many years a professor at the University of Michigan.. He is known for Lyndon words, the Curtis–Hedlund–Lyndon theorem, Craig–Lyndon interpolation and the Lyndon–Hochschild–Serre spectral sequence.

Biography

Lyndon was born on December 18, 1917, in Calais, Maine, the son of a Unitarian minister. His mother died when he was two years old, after which he and his father moved several times to towns in Massachusetts and

New York New York most commonly refers to: * New York City, the most populous city in the United States, located in the state of New York * New York (state), a state in the northeastern United States New York may also refer to: Film and television * '' ...

. He did his undergraduate studies at Harvard University, originally intending to study literature but eventually settling on mathematics, and graduated in 1939. He took a job as a banker, but soon afterwards returned to graduate school at Harvard, earning a master's degree in 1941. After a brief teaching stint at the

Georgia Institute of Technology The Georgia Institute of Technology, commonly referred to as Georgia Tech or, in the state of Georgia, as Tech or The Institute, is a public research university and institute of technology in Atlanta, Georgia. Established in 1885, it is part of ...

, he returned to Harvard for the third time in 1942 and while there taught navigation as part of the V-12 Navy College Training Program while earning his Ph.D. He received his doctorate in 1946 under the supervision of Saunders Mac Lane.. After graduating from Harvard, Lyndon worked at the

Office of Naval Research The Office of Naval Research (ONR) is an organization within the United States Department of the Navy responsible for the science and technology programs of the U.S. Navy and Marine Corps. Established by Congress in 1946, its mission is to plan ...

and then for five years as an instructor and assistant professor at Princeton University before moving to the University of Michigan in 1953. At Michigan, he shared an office with Donald G. Higman; his notable doctoral students there included

Kenneth Appel Kenneth Ira Appel (October 8, 1932 – April 19, 2013) was an American mathematician who in 1976, with colleague Wolfgang Haken at the University of Illinois at Urbana–Champaign, solved one of the most famous problems in mathematics, the four-c ...

and

Joseph Kruskal Joseph Bernard Kruskal, Jr. (; January 29, 1928 – September 19, 2010) was an American mathematician, statistician, computer scientist and psychometrician. Personal life Kruskal was born to a Jewish family in New York City to a successful fur ...

. Lyndon died on June 8, 1988, in

Ann Arbor, Michigan Ann Arbor is a city in the U.S. state of Michigan and the county seat of Washtenaw County, Michigan, Washtenaw County. The 2020 United States census, 2020 census recorded its population to be 123,851. It is the principal city of the Ann Arbor ...

Research

Lyndon's Ph.D. thesis concerned

group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology loo ...

; the Lyndon–Hochschild–Serre spectral sequence, coming out of that work, relates a group's cohomology to the cohomologies of its

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

s and their

quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...

s. A Lyndon word is a nonempty

string String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...

of symbols that is smaller, lexicographically, than any of its cyclic rotations; Lyndon introduced these words in 1954 while studying the bases of free groups. Lyndon was credited by

Gustav A. Hedlund Gustav Arnold Hedlund (May 7, 1904 – March 15, 1993), an American mathematician, was one of the founders of symbolic dynamics, symbolic and topological dynamics. Biography Hedlund was born May 7, 1904, in Somerville, Massachusetts. He did his ...

for his role in the discovery of the Curtis–Hedlund–Lyndon theorem, a mathematical characterization of cellular automata in terms of

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

equivariant In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...

functions on

shift space In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words that represent the evolution of a discrete system. In fact, shift spaces and '' symbolic dynamical systems'' are often considered synon ...

s. The Craig–Lyndon interpolation theorem in formal logic states that every logical implication can be factored into the composition of two implications, such that each nonlogical symbol in the middle formula of the composition is also used in both of the other two formulas. A version of the theorem was proved by William Craig in 1957, and strengthened by Lyndon in 1959. In addition to these results, Lyndon made important contributions to combinatorial group theory, the study of

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

in terms of their presentations in terms of sequences of generating elements that combine to form the group identity.

Awards and honors

The book ''Contributions to Group Theory'' (American Mathematical Society, 1984, ) is a festschrift dedicated to Lyndon on the occasion of his 65th birthday; it includes five articles about Lyndon and his mathematical research, as well as 27 invited and refereed research articles. The Roger Lyndon Collegiate Professorship of Mathematics at the University of Michigan, held by Hyman Bass in 1999–2008, is named after Lyndon.

Publications

Lyndon was the author or coauthor of the books: *''Notes on Logic'' (Van Nostrand, 1967) *''Word Problems: Decision Problem in Group Theory'' (with W. W. Boone and F.B. Cannonito, North-Holland, 1973) *''Combinatorial Group Theory'' (with

Paul Schupp Paul Eugene Schupp (born March 12, 1937, died January 24, 2022) was a professor emeritus of mathematics at the University of Illinois at Urbana Champaign. He is known for his contributions to geometric group theory, computational complexity and ...

, 1976, reprinted 2001 by Springer-Verlag, ) *''Groups and Geometry'' (Cambridge University Press, 1985, ). Some of his most cited papers include: * *

References

{{DEFAULTSORT:Lyndon, Roger Conant 1917 births 1988 deaths 20th-century American mathematicians Harvard University alumni Georgia Tech faculty Princeton University faculty University of Michigan faculty People from Calais, Maine Mathematicians from Maine Group theorists Academics from Maine