Samuel Eilenberg | |
---|---|

Born | |

Died | January 30, 1998 84) New York City, United States | (aged

Citizenship | Russian, Polish, American |

Alma mater | University of Warsaw |

Known for | Eilenberg–Steenrod axioms Eilenberg swindle |

Awards | Wolf Prize (1986) Leroy P. Steele Prize (1987) |

Scientific career | |

Fields | Mathematics |

Institutions | Columbia University |

Thesis | On the Topological Applications of Maps onto a Circle (1936) |

Doctoral advisors | Kazimierz Kuratowski Karol Borsuk |

Doctoral students | Jonathan Beck David Buchsbaum Martin Golumbic Daniel Kan William Lawvere Ramaiyengar Sridharan Myles Tierney |

**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.

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.^{ [1] } He died in New York City in January 1998.

Eilenberg's main body of work was in algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (whose names the Eilenberg–Steenrod axioms bear), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory.

Eilenberg was a member of Bourbaki and, with Henri Cartan, wrote the 1956 book *Homological Algebra*.^{ [2] }

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.

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*.^{ [3] } In reciprocity, the Metropolitan Museum of Art donated substantially to the endowment of the Samuel Eilenberg Visiting Professorship in Mathematics at Columbia University.^{ [4] }

- Eilenberg, Samuel (1974).
*Automata, Languages and Machines, Volume A*. ISBN 0-12-234001-9. - Eilenberg, Samuel (1976).
*Automata, Languages and Machines, Volume B*. 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.^{ [5] }

- ↑ Samuel Eilenberg at the Mathematics Genealogy Project
- ↑ Mac Lane, Saunders (1956). "Review:
*Homological algebra*, by Henri Cartan and Samuel Eilenberg".*Bulletin of the American Mathematical Society*.**62**(6): 615–624. doi: 10.1090/S0002-9904-1956-10082-7 . - ↑ Pace, Eric (February 3, 1998), "Samuel Eilenberg, 84, Dies; Mathematician at Columbia",
*The New York Times* - ↑ Bass, Hyman; Cartan, Henri; Freyd, Peter; Heller, Alex; Mac Lane, Saunders (1998). "Samuel Eilenberg (1913–1998)" (PDF).
*Notices of the American Mathematical Society*.**45**(10): 1344–1352. - ↑ Spanier, Edwin H. (1958). "Review:
*Foundations of Algebraic Topology*, by S. Eilenberg and N. Steenrod".*Bulletin of the American Mathematical Society*.**64**(4): 190–192. doi: 10.1090/s0002-9904-1958-10204-9 .

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**Henri Paul Cartan** was a French mathematician who made substantial contributions to algebraic topology. He was the son of the French mathematician Élie Cartan and the brother of composer Jean Cartan.

In mathematics, particularly algebraic topology and homology theory, the **Mayer–Vietoris sequence** is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.

**Norman Earl Steenrod** was an American mathematician most widely known for his contributions to the field of algebraic topology.

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In mathematics, **abstract nonsense**, **general abstract nonsense**, **generalized abstract nonsense**, and **general nonsense** are terms used by mathematicians to describe abstract methods related to category theory and homological algebra. More generally, “abstract nonsense” may refer to a proof that relies on category-theoretic methods, or even to the study of category theory itself.

In mathematics, the **Tor functors** are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor_{1} and the torsion subgroup of an abelian group.

In mathematics, specifically algebraic topology, an **Eilenberg–MacLane space** is a topological space with a single nontrivial homotopy group. As such, an Eilenberg–MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory; general topological spaces can be constructed from these via the Postnikov system. These spaces are important in many contexts in algebraic topology, including constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

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In mathematics, specifically in algebraic topology, the **Eilenberg–Steenrod axioms** are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod.

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In mathematics, the **standard complex**, also called **standard resolution**, **bar resolution**, **bar complex**, **bar construction**, is a way of constructing resolutions in homological algebra. It was first introduced for the special case of algebras over a commutative ring by Samuel Eilenberg and Saunders Mac Lane (1953) and Henri Cartan and Eilenberg (1956, IX.6) and has since been generalized in many ways.

The article "**Sur quelques points d'algèbre homologique**" by Alexander Grothendieck, now often referred to as the ** Tôhoku paper**, was published in 1957 in the

**Sze-Tsen Hu**, also known as **Steve Hu**, was a Chinese-American mathematician, specializing in homotopy theory.

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