![Lyndon](https://upload.wikimedia.org/wikipedia/commons/e/ea/Lyndon.jpeg)
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, structure, space, models, and change.
History
On ...
, 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 in ...
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 cro ...
, 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
Massachusetts (Massachusett: ''Muhsachuweesut Massachusett_writing_systems.html" ;"title="nowiki/> məhswatʃəwiːsət.html" ;"title="Massachusett writing systems">məhswatʃəwiːsət">Massachusett writing systems">məhswatʃəwiːsət'' En ...
and
New York. 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, 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 Taftville ...
.
[.]
After graduating from Harvard, Lyndon worked at the
Office of Naval Research 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 ...
before moving to 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 ...
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 fou ...
and
Joseph Kruskal.
Lyndon died on June 8, 1988, in
Ann Arbor, Michigan.
Research
Lyndon's Ph.D. thesis concerned
group cohomology;
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' ...
, coming out of that work, relates a group's cohomology to the cohomologies of its
normal subgroups and their
quotient groups.
A
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 in ...
is a nonempty
string 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 a ...
, 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''−1' ...
s.
Lyndon was credited by
Gustav A. Hedlund for his role in the discovery of 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 ...
, 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 ...
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 grou ...
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
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 premis ...
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. Present ...
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 h ...
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). The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the c ...
References
External links
*Directory page at University of MichiganAuthor profilein the database zbMATH
{{DEFAUL ...
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, 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