Roger Lyndon
Encyclopedia
Roger Conant Lyndon was an American mathematician, for many years a professor at the University of Michigan
. He is known for Lyndon word
s, the Curtis–Hedlund–Lyndon theorem
, Craig–Lyndon interpolation
and the Lyndon–Hochschild–Serre spectral sequence
.
, 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. 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 masters 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
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
and then for five years as an instructor and assistant professor at Princeton University
before moving to Michigan in 1953. At Michigan, he shared an office with Donald G. Higman
; his notable doctoral students there included Kenneth Appel
and Joseph Kruskal
.
Lyndon died on June 8, 1988, in Ann Arbor, Michigan
.
; the Lyndon–Hochschild–Serre spectral sequence
, coming out of that work, relates a group's cohomology to the cohomologies of its normal subgroup
s and their quotient group
s.
A Lyndon word
is a string
of symbols that is smaller, lexicographically, than any of its cyclic rotations; Lyndon introduced these words in 1954 while studying the bases of free group
s.
Lyndon was credited by Gustav A. Hedlund
for his role in the discovery of the Curtis–Hedlund–Lyndon theorem
, a mathematical characterization of cellular automata
in terms of continuous
equivariant functions on shift space
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
in terms of their presentations
in terms of sequences of generating elements that combine to form the group identity.
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
since 1999, is named after Lyndon.
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
. He is known for Lyndon word
Lyndon word
In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a string that is strictly smaller in lexicographic order than all of its rotations...
s, the Curtis–Hedlund–Lyndon theorem
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...
, Craig–Lyndon interpolation
Craig interpolation
In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ then there is a third formula ρ, called an interpolant, such that every nonlogical symbol...
and the Lyndon–Hochschild–Serre spectral sequence
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 and the quotient group G/N to the cohomology of the...
.
Biography
Lyndon was born on December 18, 1917 in Calais, MaineCalais, Maine
Calais is a city in Washington County, Maine, United States. The city has three United States border crossings or also known as a Port of entry with the busiest being on the St. Croix River bordering St. Stephen, New Brunswick, Canada...
, the son of a Unitarian
Unitarianism
Unitarianism is a Christian theological movement, named for its understanding of God as one person, in direct contrast to Trinitarianism which defines God as three persons coexisting consubstantially as one in being....
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. He did his undergraduate studies at Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
, 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 masters degree in 1941. After a brief teaching stint at the Georgia Institute of Technology
Georgia Institute of Technology
The Georgia Institute of Technology is a public research university in Atlanta, Georgia, in the United States...
, 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
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...
while earning his Ph.D. He received his doctorate in 1946 under the supervision of Saunders Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
.
After graduating from Harvard, Lyndon worked at the Office of Naval Research
Office of Naval Research
The Office of Naval Research , headquartered in Arlington, Virginia , is the office within the United States Department of the Navy that coordinates, executes, and promotes the science and technology programs of the U.S...
and then for five years as an instructor and assistant professor at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
before moving to Michigan in 1953. At Michigan, he shared an office with Donald G. Higman
Donald G. Higman
Donald G. Higman was an American mathematician known for his discovery, in collaboration with Charles C. Sims, of the Higman–Sims group....
; his notable doctoral students there included Kenneth Appel
Kenneth Appel
Kenneth Ira Appel is a 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-color theorem...
and Joseph Kruskal
Joseph Kruskal
Joseph Bernard Kruskal, Jr. was an American mathematician, statistician, computer scientist and psychometrician. He was a student at the University of Chicago and at Princeton University, where he completed his Ph.D. in 1954, nominally under Albert W...
.
Lyndon died on June 8, 1988, in Ann Arbor, Michigan
Ann Arbor, Michigan
Ann Arbor is a city in the U.S. state of Michigan and the county seat of Washtenaw County. The 2010 census places the population at 113,934, making it the sixth largest city in Michigan. The Ann Arbor Metropolitan Statistical Area had a population of 344,791 as of 2010...
.
Research
Lyndon's Ph.D. thesis concerned group cohomologyGroup cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
; the Lyndon–Hochschild–Serre spectral sequence
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 and the quotient group G/N to the cohomology of the...
, coming out of that work, relates a group's cohomology to the cohomologies of its normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
s and their quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
s.
A Lyndon word
Lyndon word
In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a string that is strictly smaller in lexicographic order than all of its rotations...
is a string
String (computer science)
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....
of symbols that is smaller, lexicographically, than any of its cyclic rotations; Lyndon introduced these words in 1954 while studying the bases of free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
s.
Lyndon was credited by Gustav A. Hedlund
Gustav A. Hedlund
Gustav Arnold Hedlund , 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 studies at Harvard University, earned a masters degree from Columbia University, and...
for his role in the discovery of the Curtis–Hedlund–Lyndon theorem
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...
, a mathematical characterization of cellular automata
Cellular automaton
A cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...
in terms of continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
equivariant functions on shift space
Shift space
In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words representing the evolution of a discrete system. In fact, shift spaces and symbolic dynamical systems are often considered synonyms....
s.
The Craig–Lyndon interpolation theorem
Craig interpolation
In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ then there is a third formula ρ, called an interpolant, such that every nonlogical symbol...
in formal logic
Formal logic
Classical or traditional system of determining the validity or invalidity of a conclusion deduced from two or more statements...
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
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...
, the study of groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
in terms of their presentations
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
in terms of sequences of generating elements that combine to form the group identity.
Books
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, ISBN 978-3540411581)
- Groups and Geometry (Cambridge University Press, 1985, ISBN 978-0521316941).
Awards and honors
The book Contributions to Group Theory (American Mathematical Society, 1984, ISBN 9780821850350) is a festschriftFestschrift
In academia, a Festschrift , is a book honoring a respected person, especially an academic, and presented during his or her lifetime. The term, borrowed from German, could be translated as celebration publication or celebratory writing...
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
Hyman Bass is an American mathematician, known for work in algebra and in mathematics education. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus...
since 1999, is named after Lyndon.