EILENBERG, SAMUEL

(b. Warsaw, Russian Empire [later Poland], 13 September 1913;

d. New York, New York, 30 January 1998), mathematics, specifically algebraic topology, category theory, and automata theory.

Eilenberg, one of the architects of twentieth-century mathematics, transformed mathematicians’ ways of thinking about topology and, in the process, helped found two major branches of mathematics—homological algebra and category theory—that he later applied to the theory of automata. Topology is an elastic version of geometry that retains the idea of continuity but relaxes rigid metric notions of distance. Its ideas were first introduced by Henri Poincaré in the early twentieth century, but topology became a central frontier of mathematical research only in the latter half of that century, with the introduction of powerful algebraic methods, largely pioneered by Eilenberg.

Early Life and Career . Little is known of Eilenberg’s family life in Poland. Eilenberg recounted in a conversation with Peter Freyd, “My mother’s father had the town brewery and he had one child, a daughter. He went to the head of the town Yeshiva and asked for the best student. … So my future father became a brewer instead of a rabbi” (Bass et al, p. 1350). As a personal presence Eilenberg was, throughout his life, short, energetic (for example, an enthusiastic swimmer in his youth), expressive, charismatic, quick witted, often confrontational, and a brilliant thinker and good-humored conversationalist. He was single, except for his marriage to Natasha Chterenzon from 1960 to 1969. His development as a mathematician began at the University of Warsaw, in the vibrant Polish school of general topology. After receiving his master of arts degree in 1934, Eilenberg began his thesis, concerned with the topology of the plane and written under the direction of Karol Borsuk. It was well received abroad as

well as in Poland, and Eilenberg received his PhD in 1936. He published prodigiously, largely in French, during his early years in Poland, and so he already had an international reputation when he left that nation.

Early in 1939, on the advice of his father, Eilenberg left Poland for New York. Soon after his arrival he went to Princeton, New Jersey, where Oswald Veblen and Solomon Lefschetz of Princeton University were welcoming refugee mathematicians and finding them suitable positions at American universities. By this time, Eilenberg had already come to be called “Sammy” by all mathematicians who knew him. He joined the Topology Group then being developed by Ray Wilder at the University of Michigan, which came to include such major figures as Norman Steenrod, Deane Montgomery, Hans Samelson, Raoul Bott, and Steven Smale.

At Michigan: . Eilenberg and Steenrod . At the University of Michigan began Eilenberg’s collaboration with Steenrod, which led to their axiomatization of homology theory and to their landmark book, Foundations of Algebraic Topology(1952), which synthesized and crystallized the then-chaotic state of the field. Eilenberg’s contributions to this work earned him the Leroy P. Steele Prize of the American Mathematical Society in 1987. Axiomatization afforded important advantages. Verification of the axioms became an elegantly efficient way to establish the equivalence of different definitions of homology. Moreover, the axioms and variations on them provided a general and uniform framework for mathematical development, the results of which apply to all models of the axioms, often well beyond those that first motivated their introduction.

Such a program of development was vigorously pursued in what, after this Eilenberg-Steenrod publication, came to be called Algebraic Topology. In building up the vast algebraic machinery for the development of algebraic topology, Eilenberg began long-term collaborations with Henri Cartan of the Université de Paris and with Saunders Mac Lane of the University of Chicago. Though his mathematical ideas may have seemed to have a kind of crystalline austerity, Eilenberg was a warm, robust, and very animated human being, for whom mathematics was a social activity—whence his many collaborations. He liked to do mathematics on his feet, often prancing while he explained his thoughts. When something connected, one could read it in his impish smile and the sparkle in his eyes.

At Columbia: . Eilenberg and Cartan, and Bourbaki .

Eilenberg first met Cartan in 1947 in New York City, when he began his thirty-five-year career at Columbia University. Eilenberg also later lectured in Paris in the “Séminaire Cartar” (Cartan Seminar) at the École normale supérieure (Superior normal school), then attended by the students Jean-Pierre Serre and Armand Borel. His collaboration with Cartan evolved into the epochal work, Homological Algebra (1956). This book identified and systematized the fundamental algebraic structures that supported the new Algebraic Topology but put them on an autonomous algebraic footing—a grand work of distillation and synthesis. The tools and language of homological algebra later infiltrated, often in deep ways, every branch of mathematics.

Eilenberg’s French connection went deeper. In 1949, André Weil invited Eilenberg to participate in what was called Bourbaki, an illustrious but anonymous group of French mathematicians who were writing, under the pseudonym of Nicolas Bourbaki, a multivolume tome aimed at providing a coherent and rigorous treatment of the fundamental structures of all of contemporary mathematics. Eilenberg, as one of the rare non-French members, collaborated actively with Bourbaki for the next fifteen years.

Eilenberg and Mac Lane . Eilenberg’s other great collaboration, with Saunders Mac Lane, began when, in 1940, the latter came to lecture in Michigan on “group extensions,” a fundamental construction in the theory of groups, which is the mathematical theory of symmetry. Eilenberg immediately saw connections that answered a problem that had been raised by Steenrod. Eilenberg and Mac Lane stayed up the night working out details of what later evolved into a major series of papers that founded the cohomology theory of groups, among other things. (This research was done partly in the spare hours of wartime work on the mathematics of ballistics.) Much of the early work of Eilenberg and Mac Lane, beyond its important contributions to algebraic topology, served as a precursor to Homological Algebra.

Most significantly, Mac Lane understood and came to embrace Eilenberg’s mathematical spirit and method, which was to look always for fundamental explanatory structures, unencumbered by anything extraneous, that made the mathematics flow naturally, almost inevitably, in a way that demystified the complex. Eilenberg believed philosophically that all mathematics, once properly understood, would submit to such treatment. Mac Lane captured part of Eilenberg’s spirit with the maxim, “Dig deep and deeper, till you get to the bottom of each issue.” Eilenberg’s student, Alex Heller, expressed it as “his radical insistence on lucidity, order, and understanding as opposed to trophy hunting.” Peter Freyd, another of Eilenberg’s students, expressed it more lyrically as a “triumph of style over substance” (all quotations from Bass et al.) Eilenberg’s honorary degree from the University of Pennsylvania in 1985 cited him as “our greatest mathematical stylist.”

Eilenberg’s Mathematical Legacy . In the words of Eilenberg’s Columbia colleague, John Morgan, “The theme that runs through Sammy’s mathematics is always to find the absolutely essential ingredients in any problem and work only with those ingredients and nothing else—in other words, get rid of all the superfluous information.” For example, when someone once asked Eilenberg if he could eat Chinese food with three chopsticks, he answered, “Of course.” The questioner then asked, “How are you going to do it?” and Eilenberg replied, “I’ll take the three chopsticks, I’ll put one of them aside on the table, and I’ll use the other two” (Pace, p. B9).

The greatest monument of Eilenberg and Mac Lane’s work is the field of category theory, substantially created by them and their students and disciples. While some of the seeds of category theory were already visible in Homo-logical Algebra, it emerges as a kind of reification of the formalisms of all of mathematics, in terms of “matter” (objects) and “motion” (transformations, or arrows). What is remarkable, at this extreme level of abstraction and voiding of internal meaning, is how much interesting and useful mathematics remains to be done, and discovered. Indeed, the innocuous-seeming 1945 paper of Eilenberg and Mac Lane, “General Theory of Natural Equivalences,” in which categories were first defined, was first rejected by the editor of an inauspicious journal as “more devoid of content than any I have read.” To which Mac Lane is said to have replied, “That’s the point.” Besides becoming, in the hands of Eilenberg’s students— notably William Lawvere—a vigorous field in its own right, category theory also helped shape Alexandre Grothendieck’s refounding of modern algebraic geometry, and it has had significant applications in logic, theoretical computer science, linguistics, and philosophy. Eilenberg and Mac Lane continued throughout their careers to nurture the lively international community of category theorists, for which they became venerated and generous patriarchs. The fifteen joint papers of Eilenberg and Mac Lane are assembled in Eilenberg-Mac Lane, Collected Works (1986).

Eilenberg also adopted a categorical approach to the last phase of his mathematical work, which was in automata theory, a branch of theoretical computer science. This resulted in the two-volume work, Automata, Languages, and Machines(1974–1976).

Eilenberg and Columbia University . Eilenberg spent most of his mathematical career in the Mathematics Department of Columbia University, which he twice chaired. He helped develop it into a major center of pure mathematical research. Beyond his many doctoral students, including David Buchsbaum, Peter Freyd, Alex Heller, Daniel Kan, William Lawvere, Fred Linton, and Steven Schanuel, he was an important mentor to many of the postdoctoral fellows at Columbia, this author included. In 1982, Columbia named Eilenberg a university professor, the highest faculty distinction that the university confers.

Eilenberg was a member of the National Academy of Sciences (U.S.A.) and the American Academy of Arts and Sciences. In 1986, Eilenberg shared the Wolf Prize in mathematics, “for his fundamental work in algebraic topology and homological algebra.” His co-awardee was the number theorist, Atle Selberg.

Art Collector and Dealer . Though a celebrated and well-liked mathematician on the world stage, Eilenberg also lived in another world, separate and almost parallel to that of mathematics, but intersecting it, by design, only near the end of his career. In one world lived “Sammy,” the mathematician; in the other world lived the “Professor,” famous as a dealer in and collector of art, a world where few who knew the “Professor” knew that he was also a celebrated mathematician. The “Professor's” interest was specifically Southeast Asian sculpture, of which Eilenberg gradually accumulated a rare and valuable collection. His collecting began on excursions during mathematical visits to India in the mid-1950s. He found in ancient Hindu sculpture a formal elegance and imagination that resonated well with the same aesthetic sensibility—“classical rather than romantic,” in the words of Alex Heller—that animated his mathematical work.

Not being particularly wealthy, he chose a niche that leveraged his modest material resources, refined aesthetic judgment, and acute skills as a dealer to best advantage. He generally gathered small pieces from regions and periods that were not yet saturated by the current art market. He often found his way to primary sources and became a world expert—widely consulted by dealers and curators— in the genre, on both authentic and counterfeit pieces. His collection of Javanese bronzes was said to be the most important group outside Indonesia and the Netherlands. After thirty years of collecting, his collection was estimated in 1989 to be worth more than $5 million. Parts of his collection have been exhibited in major museums— the Metropolitan Museum of Art in New York City, the Arthur M. Sackler Gallery in Washington, D.C., the Dallas Museum of Art, the Cleveland Museum of Art, the Brooklyn Museum, and the British Museum in London, as well as the Victoria and Albert Museum in London. Mathematician friends of Eilenberg could glimpse some of the gems of his collections in his apartments, one on Riverside Drive in New York City, the other in London, near Buckingham Palace. They seemed among the principal occupants of these templelike living spaces.

Eilenberg’s art and mathematical worlds intersected when, in 1989, he donated more than four hundred valuable sculptures to the Metropolitan Museum of Art. In turn, the Metropolitan raised, through general funds and with contributions by others, most of the $1.5 million needed to endow the Samuel Eilenberg Visiting Professorship of Mathematics at Columbia University. This elegant maneuver, optimizing outcomes for all concerned, was a vintage Eilenberg design. The Eilenberg Visiting Professorship has brought many of the world’s leading mathematicians to Columbia.

Eilenberg’s aesthetic sensibility also valued the artisanship and imagination of more mundane objects, as shown in another collection he assembled, this time of Indian betel nut cutters. These are often ornate hinged devices, typically made of brass, for cutting the hard betel nut that is commonly chewed in South Asia. The forms and ornamentation of them can be extravagantly imaginative and expressive. Previously little recognized in the art world, Eilenberg’s collection, described in Henry Brownrigg’s Betel Cutters from the Samuel Eilenberg Collection (1992), helped create a niche for these objects in the art collectors’ world.

Eilenberg led a full and active life till, in 1995, in New York City, he suffered a stroke. He remained mentally alert but was bedridden; sadly, he lost his ability to speak. His health remained frail. In June 1997, he fell into a coma, a state in which he lingered until his death of cardiac arrest at a geriatric center in New York City in January 1998, at the age of eighty-four.

BIBLIOGRAPHY

WORKS BY EILENBERG

With Saunders Mac Lane. “General Theory of Natural Equivalences.” Transactions of the American Mathematical Society58 (1945): 231–294. This modest paper is generally considered the birthplace of category theory.

With Norman Steenrod. Foundations of Algebraic Topology. Princeton, NJ: Princeton University Press, 1952.

With Henri Cartan. Homological Algebra. Princeton, NJ: Princeton University Press, 1956. Translated into Spanish and Russian.

With Calvin C. Elgot. Recursiveness. New York: Academic Press, 1970.

Automata, Languages, and Machines. 2 vols. New York, Academic Press, 1974–1976.

With Saunders Mac Lane. Eilenberg-Mac Lane, Collected Works. Orlando, FL: Academic Press, 1986.

OTHER SOURCES

Bass, Hyman, et al. “Samuel Eilenberg (1913–1998).” Notices of the American Mathematical Society 45, no. 10 (1998): 1344–1352. An extended memorial article.

Brownrigg, Henry. Betel Cutters: From the Samuel Eilenberg Collection. London; New York: Thames and Hudson, 1992.

Lerner, Martin, and Steven Kossak. The Lotus Transcendent: Indian and Southeast Asian Art from the Samuel Eilenberg Collection. New York: Metropolitan Museum of Art, 1991.

Pace, Eric. “Samuel Eilenberg, 84, Dies; Mathematician at Columbia.” New York Times, 3 February 1998, p. B9.

Hyman Bass