Roger Conant Lyndon (December 18, 1917 – June 8, 1988) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

, for many years a professor at the

University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...

.. He is known for

Lyndon word In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a nonempty string that is strictly smaller in lexicographic order than all of its rotations. Lyndon words are named after mathematician Roger Lyndon, who invest ...

s, the

Curtis–Hedlund–Lyndon theorem The Curtis–Hedlund–Lyndon theorem is a mathematical characterization of cellular automata in terms of their symbolic dynamics. It is named after Morton L. Curtis, Gustav A. Hedlund, and Roger Lyndon; in his 1969 paper stating the theorem, Hedl ...

, Craig–Lyndon interpolation and the

Lyndon–Hochschild–Serre spectral sequence In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup ''N ...

Biography

Lyndon was born on December 18, 1917, in

Calais, Maine Calais is a city in Washington County, Maine, United States. As of the 2020 census, it had a population of 3,079, making Calais the third least-populous city in Maine (after Hallowell and Eastport). The city has three Canada–US border cr ...

, the son of a

Unitarian Unitarian or Unitarianism may refer to: Christian and Christian-derived theologies A Unitarian is a follower of, or a member of an organisation that follows, any of several theologies referred to as Unitarianism: * Unitarianism (1565–present ...

minister. His mother died when he was two years old, after which he and his father moved several times to towns in

Massachusetts Massachusetts (Massachusett language, Massachusett: ''Muhsachuweesut assachusett writing systems, məhswatʃəwiːsət'' English: , ), officially the Commonwealth of Massachusetts, is the most populous U.S. state, state in the New England ...

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 Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...

, 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 The V-12 Navy College Training Program was designed to supplement the force of commissioned officers in the United States Navy during World War II. Between July 1, 1943, and June 30, 1946, more than 125,000 participants were enrolled in 131 colleg ...

while earning his Ph.D. He received his doctorate in 1946 under the supervision of

Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...

.. 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 pla ...

and then for five years as an instructor and assistant professor at

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...

before moving to the

in 1953. At Michigan, he shared an office with

Donald G. Higman Donald G. Higman (September 20, 1928 in Vancouver – February 13, 2006) was an American mathematician known for his discovery, in collaboration with Charles C. Sims, of the Higman–Sims group.. Higman did his undergraduate studies at the Un ...

; 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 fou ...

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 fu ...

. Lyndon died on June 8, 1988, in Ann Arbor, Michigan.

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 cohomolog ...

; the

, 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 ...

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 exam ...

s. A

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 In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of ...

, than any of its cyclic rotations; Lyndon introduced these words in 1954 while studying the bases of

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...

s. 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 and topological dynamics. Biography Hedlund was born May 7, 1904, in Somerville, Massachusetts. He did his undergraduate studi ...

for his role in the discovery of the

, a mathematical characterization of

cellular automata A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...

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 g ...

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 gro ...

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 synony ...

s. The Craig–Lyndon interpolation theorem in

formal logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

states that every

logical implication Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...

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 In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a nat ...

, 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 ide ...

in terms of their

presentations A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...

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 In academia, a ''Festschrift'' (; plural, ''Festschriften'' ) is a book honoring a respected person, especially an academic, and presented during their lifetime. It generally takes the form of an edited volume, containing contributions from the ...

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 Hyman Bass (; born October 5, 1932)

W. W. Boone and F.B. Cannonito, North-Holland, 1973) *''Combinatorial Group Theory'' (with Paul Schupp, 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