mathematics

Samuel Eilenberg

Samuel Eilenberg is a mathematics 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 Samuel Eilenberg rather than just read about it. In short: Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a Jewish family.

Key takeaways

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

Reference excerpt

Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.

Early life and education He was born in Warsaw, Kingdom of Poland to a Jewish family. He spent much of his career as a professor at Columbia University. He earned his Ph.D. from University of Warsaw in 1936, with thesis On the Topological Applications of Maps onto a Circle; his thesis advisors were Kazimierz Kuratowski and Karol Borsuk. He died in New York City in January 1998.

Career Eilenberg's main body of work was in algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Earl Steenrod (and the Eilenberg–Steenrod axioms are named for the pair), and on homological algebra with Saunders Mac Lane. As a result of this work, Eilenberg and Mac Lane developed the field of category theory, for which they are now best known. Eilenberg was a member of Bourbaki and, with Henri Cartan, wrote the 1956 book Homological Algebra. Later in life he worked mainly in pure category theory, being one of the founders of the field. The Eilenberg swindle (or telescope) is a construction applying the telescoping cancellation idea to projective modules. Eilenberg contributed to automata theory and algebraic automata theory. In particular, he introduced a model of computation called X-machine and a new prime decomposition algorithm for finite state machines in the vein of Krohn–Rhodes theory. He also identified a natural correspondence between certain classes of regular languages called varieties and pseudovarieties of finite monoids, a result which is now known as Eilenberg's theorem.

Art collection Eilenberg was also a prominent collector of Asian art. His collection mainly consisted of small sculptures and other artifacts from India, Indonesia, Nepal, Thailand, Cambodia, Sri Lanka and Central Asia. In 1991–1992, the Metropolitan Museum of Art in New York staged an exhibition from more than 400 items that Eilenberg had donated to the museum, entitled The Lotus Transcendent: Indian and Southeast Asian Art From the Samuel Eilenberg Collection. In reciprocity, the Metropolitan Museum of Art donated substantially to the endowment of the Samuel Eilenberg Visiting Professorship in Mathematics at Columbia University.

Selected publications

Eilenberg, Samuel (1974). Automata, Languages and Machines, Volume A. Academic Press. ISBN 0-12-234001-9. Eilenberg, Samuel (1976). Automata, Languages and Machines, Volume B. Academic Press. ISBN 0-12-234002-7. Eilenberg, Samuel; Ganea, Tudor (1957). "On the Lusternik-Schnirelmann category of abstract groups". Annals of Mathematics. 2nd Series. 65 (3): 517–518. doi:10.2307/1970062. JSTOR 1970062. MR 0085510. Eilenberg, Samuel; Mac Lane, Saunders (1945). "Relations between homology and homotopy groups of spaces". Annals of Mathematics. 46 (3): 480–509. doi:10.2307/1969165. JSTOR 1969165. Eilenberg, Samuel; Mac Lane, Saunders (1950). "Relations between homology and homotopy groups of spaces. II". Annals of Mathematics. 51 (3): 514–533. doi:10.2307/1969365. JSTOR 1969365. Eilenberg, Samuel; Moore, John C. (1962), "Limits and spectral sequences", Topology, 1 (1): 1–23, doi:10.1016/0040-9383(62)90093-9, ISSN 0040-9383 Eilenberg, Samuel; Niven, Ivan (1944). "The "fundamental theorem of algebra" for quaternions". Bulletin of the American Mathematical Society. 50 (4): 246–248. doi:10.1090/s0002-9904-1944-08125-1. MR 0009588. Eilenberg, Samuel; Steenrod, Norman E. (1945). "Axiomatic approach to homology theory". Proceedings of the National Academy of Sciences of the United States of America. 31 (4): 117–120. Bibcode:1945PNAS...31..117E. doi:10.1073/pnas.31.4.117. PMC 1078770. PMID 16578143. Eilenberg, Samuel; Steenrod, Norman E. (1952). Foundations of Algebraic Topology. Princeton, New Jersey: Princeton University Press. MR 0050886.

See also Stefan Banach Stanislaw Ulam Eilenberg–Montgomery fixed point theorem

Footnotes

External links O'Connor, John J.; Robertson, Edmund F., "Samuel Eilenberg", MacTutor History of Mathematics Archive, University of St Andrews Eilenberg's biography − from the National Academies Press, by Hyman Bass, Henri Cartan, Peter Freyd, Alex Heller and Saunders Mac Lane.

Illustrations

Worked examples

Example 1 — a first encounter with Samuel Eilenberg

Start with the simplest possible case. Write down what Samuel Eilenberg claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Samuel Eilenberg 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 Samuel Eilenberg 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 Samuel Eilenberg

In research: Samuel Eilenberg appears in mathematics 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 Samuel Eilenberg 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: Samuel Eilenberg is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1913 births, 1998 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Samuel Eilenberg 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 Samuel Eilenberg in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Samuel Eilenberg 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 Samuel Eilenberg out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Samuel Eilenberg in simple terms?

Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a Jewish family.

Why does Samuel Eilenberg matter?

Because it connects several mathematics 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 Samuel Eilenberg?

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 Samuel Eilenberg.