astronomy

Roger Lyndon

Roger Lyndon is a astronomy topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Roger Lyndon rather than just read about it. In short: 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.

Key takeaways

Roger Lyndon belongs to astronomy; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Roger Lyndon to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Roger Lyndon from memory before moving on to harder problems.

Reference excerpt

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. 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, 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 the University of 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.

Research Lyndon's Ph.D. thesis concerned group cohomology; the Lyndon–Hochschild–Serre spectral sequence, coming out of that work, relates a group's cohomology to the cohomologies of its normal subgroups and their quotient groups. A Lyndon word is a nonempty 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 groups. 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 spaces. 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.

Awards and honors The book Contributions to Group Theory (American Mathematical Society, 1984, ISBN 978-0-8218-5035-0) 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, 1976, reprinted 2001 by Springer-Verlag, ISBN 978-3-540-41158-1) Groups and Geometry (Cambridge University Press, 1985, ISBN 978-0-521-31694-1). Some of his most cited papers include:

Lyndon, Roger C. (1950). "Cohomology theory of groups with a single defining relation". Annals of Mathematics. 52 (3): 650–665. doi:10.2307/1969440. JSTOR 1969440. MR 0047046. Chen, Kuo Tsai; Fox, Ralph H.; Lyndon, Roger C. (1958). "Free differential calculus. IV. The quotient groups of the lower central series". Annals of Mathematics. 68 (1): 81–95. doi:10.2307/1970044. JSTOR 1970044. MR 0102539.

References

Illustrations

Roger Lyndon: Roger Lyndon — Roger Lyndon

Worked examples

Example 1 — a first encounter with Roger Lyndon

Start with the simplest possible case. Write down what Roger Lyndon claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In astronomy, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Roger Lyndon before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Roger Lyndon ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Roger Lyndon

In research: Roger Lyndon appears in astronomy research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Roger Lyndon in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Roger Lyndon is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1917 births, 1988 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Roger Lyndon outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Roger Lyndon in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Roger Lyndon means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Roger Lyndon out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Roger Lyndon in simple terms?

Why does Roger Lyndon matter?

Because it connects several astronomy ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Roger Lyndon?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Roger Lyndon.