**David Eisenbud**'s parents were Leonard Eisenbud and Ruth-Jean Rubinstein. Leonard Eisenbud's parents, Boris and Katherine Eisenbud, were Russian but had emigrated to the United States in 1902. Leonard Eisenbud (1913-2004) was a student of Eugene Wigner and together they co-authored the important book *Nuclear Structure* (1958). During World War II, Leonard had undertaken research on radar, and with Wigner, had lectured on quantum mechanics at the Clinton Laboratories at Oak Ridge. In 1947, shortly after their son David was born, the Eisenbuds moved to a newly purchased house in Patchogue after accepting the offer to join the new Brookhaven National Laboratory's Physics Department situated on Long Island, Upton, New York. Francis Bonner describes what happened next [2]:-

I hadn't been there very long when I was shocked to learn that Leonard Eisenbud's "Q" clearance was in jeopardy: he had received an "interrogatory" asking him to respond to a broad range of questions about his and his wife's political affiliations. It was no secret that Ruth-Jean's mother and sister were members of the Communist Party, and the charges against Leonard began and wandered off from there. Indignant and dispirited, and believing the probability of his receiving clearance to be vanishingly small, Leonard chose not to pursue it to the stage of formal hearings and left to accept an appointment at the Bartol Research Foundation in Swarthmore, PA.

David Eisenbud was brought up in Swarthmore where his father worked until 1958 when the State University of New York set up the State University College on Long Island at Oyster Bay, and Leonard Eisenbud was appointed Professor of Physics and Acting Chairman of the Department of Physics. So far we have not mentioned the career of David's mother Ruth-Jean Eisenbud (1915-2004) - we should note that she was a childhood polio victim but went on to become a psychotherapist and a professor at New York University and Adelphi University.

Eisenbud entered the University of Chicago where he went on to undertake both undergraduate and graduate studies. He was only nineteen years old when he graduated with a B.S. in 1966 and continued to graduate studies, being awarded his Master's degree in 1967. He then undertook research with his thesis advisor Saunders Mac Lane but he was given considerable help by J C Robson who he considered as an unofficial advisor. He wrote of Mac Lane:-

He was a great figure, and very important for me personally.

But others at Chicago were also strong influences on the young Eisenbud [1]:-

While I was a graduate student at the University of Chicago(1967-1970), I listened at every chance I got to the beautiful lectures of Irving Kaplansky. He was then just finishing his book 'Commutative Rings', and lectured from it. I admired him and it a great deal, but - in the style of a rebellious adolescent - I was quite ready to proclaim that a lot was left out.

On 3 June 1970, Eisenbud married Monika Margarete Schwabe; they had two children David and Alina. Also in 1970, Eisenbud received his Ph.D. from Chicago for his thesis on non-commutative ring theory* Torsion Modules over Dedekind Prime Rings*. Eisenbud's first paper was not on ring theory, however, but rather on group theory with *Groups of order automorphisms of certain homogeneous ordered sets* (1969). In 1970 he published a number of papers on non-commutative ring theory: *Subrings of Artinian and Noetherian rings*; (with J C Robson) *Modules over Dedekind prime rings*; and (with J C Robson) *Hereditary Noetherian prime rings*. He was appointed as a lecturer at Brandeis University in 1970 and taught there for twenty-seven years. J C Robson, an English ring theorist, was at the University of Leeds in England and Eisenbud visited him in 1971 [1]:-

In the fall of1971, visiting at the University of Leeds in England, I had a chance to lecture on ... Noether normalization(in a version borrowed from Nagata's book, 'Local Rings').

He was promoted at Brandeis University to Assistant Professor in 1972. In the following year, he was awarded a Sloan Foundation Fellowship which allowed him to spend the two academic years 1973-75 on research visits. He was a visiting scholar at Harvard University during 1973-74 and then was an invited speaker at the International Congress of Mathematicians in Vancouver in August 1974. Continuing his research visits, he was a Fellow at the Institut des Hautes Etudes Scientiques (Bures-Sur-Yvette) during 1974-75:-

... thoroughly enjoying the French for their mathematics, culture, and good life. ... About half way through that year I was invited by Wolfgang Vogel, whom I had never met, to come to Halle, in East Germany, to visit, give a lecture, and discuss mathematics. ... A refugee from neighbouring Leipzig when she was five, my wife hadn't been back since then and came along. The trip seemed exotic and slightly risky. We had plenty of adventures that first visit! On the train from Leipzig to Halle my wallet - with my passport - mysteriously disappeared from my coat, a loss that we discovered as soon as I tried to check in at the University guest house.

Returning to Brandeis University, Eisenbud was promoted to Associate Professor in 1976. He was a Visiting Professor at the University of Bonn during the academic year 1979-80, then promoted to full professor at Brandeis in 1980. Barry Mazur describes his research achievements in [9]. We give some extracts:-

David Eisenbud's research accomplishments extend broadly through algebra and its applications. His publications(over a hundred of them!)have made significant contributions to fundamental issues in the subject. David also has a marvelous gift for mathematical collaboration. The sweep of his interests and the intensity of his mathematical interactions have brought him into fruitful co-authorship with many mathematicians of different backgrounds and different viewpoints.

Shortly after his graduate days, David began a joint project with Buchsbaum. Among other things, they established an elegant geometric criterion for exactness of a finite free complex that has many applications in the homological study of commutative rings. ... In the middle1970s David worked with Harold Levine on the topology of finite C[^{∞}map germs ... V I Arnold once referred tothe]celebrated formula of Eisenbud-Levine, which links calculus, algebra and geometry, as a "paradigm" more than a theorem that provides a local manifestation of interesting global invariants and that "would please Poincaré and Hilbert(also Euler, Cauchy and Kronecker, to name just those classical mathematicians, whose works went in the same direction)." Given this early work, it was natural for David's attention to turn to the study of singularities and their topology. In this period, David wrote a book with the topologist Walter Neumann[a son of Bernhard Neumann and Hanna Neumann]on the topology of the complements of the sort of knots that appear in the theory of plane-curve singularities. David next became interested in algebraic geometry, beginning a long and important collaboration with Joe Harris. Together, they developed the theory of Limit Linear Series and used it to solve a number of classical problems about the moduli spaces of complex algebraic curves.

During 1982-84, Eisenbud was Chairman of the Department of Mathematics at Brandeis University. Following this he again took research leave spending the academic year 1986-87 as a Visiting Professor at the Mathematical Sciences Research Institute at Berkeley and the following academic year as a Visiting Professor at Harvard University. Back at Brandeis, he was again Chairman of the Department of Mathematics during 1992-94. He made a research visit after this period as Chairman, spending the autumn term of 1994 at Harvard University and the spring term of 1995 at l'Institut Henri Poincaré in Paris. In 1997 he was appointed as Director of the Mathematical Sciences Research Institute (MSRI) at Berkeley and a Professor at University of California, Berkeley. Although in some sense these two posts were complementary, we should note that MSRI is not part of the University of California. This has advantages in that it gives it independence, but it also means that it is far less secure financially as it stands alone. There were many challenges and opportunities for Eisenbud taking on this new post [4]:-

MSRI is a large operation, with about1,300visitors coming through each year and about85in residence at any one time. It is also large in terms of its coverage of mathematics. Over the years it has hosted programs in mathematical economics, mathematical biology, string theory, and statistics, as well as in a wide variety of areas in pure mathematics. Indeed, Eisenbud notes that a distinctive feature of MSRI in the world of mathematics institutes is its combination of pure and applied areas. As he puts it, "We have continued to have a fundamental emphasis, and we mix it with applied areas."

However, he took up the post wanting to be far more than an administrator [7]:-

Eisenbud is looking forward not only to being the director of MSRI but also to being a mathematician there. "My own work has involved a number of different fields, and I like learning mathematics a lot," he remarks. "So I feel that I'll profit personally by the flow of mathematics through there, as well as helping the Institute."

He held this position as director for ten years and, during that time, he was also president of the American Mathematical Society from 2003 to 2005. In fact the excellent work he was doing at MSRI was a factor in his election as president. Margaret Wright writes [9]:-

David has served the mathematical community as chair of the mathematics department at Brandeis, on advisory and evaluation committees for the National Science Foundation, as a member of the Board on Mathematical Sciences, and as vice president of the AMS. But his service that is most visible nationally and internationally has been as director of MSRI, where he moved in1997after twenty-seven years at Brandeis. A fundamental strength of mathematicians is their ability to generalize, and I believe that David's performance as AMS president can be predicted with high accuracy by generalizing from his success at MSRI. In fact, his leadership at MSRI exemplifies the qualities needed by the AMS president. With David as its director, MSRI has continued its tradition of superlative programs in fundamental mathematics while simultaneously expanding into a broader and more diverse selection of fields. David has furthered a deliberate policy of outreach into new areas, and MSRI's influence and reputation increasingly extend beyond core mathematics into areas on the boundaries between mathematics and science as well as into applications ranging from imaging to cryptography to finance.

Eisenbud has published a number of important books. In 1982, in collaboration with Corrado De Concini and Claudio Procesi, he published the monograph *Hodge algebras*. Three years later, this time in collaboration with Walter Neumann, he published *Three-dimensional link theory and invariants of plane curve singularities*. In 1992, in collaboration with Joe Harris, he produced *Schemes. The language of modern algebraic geometry*. Alexey Rudakov writes in a review:-

This book is intended to introduce basic notions of modern algebraic geometry. These are schemes in general, affine schemes, projective schemes and the functor of points. The authors discuss the motivation behind most of their definitions and give many examples. The reader who works through them will become comfortable with schemes as far as flatness and characterization of a space by its functor of points are concerned, as well as with the notion itself. These topics are important in the developing field of noncommutative geometry, where scheme-theoretic thinking is involved, and this book provides a good exposition for students.

His most significant book, however, was *Commutative algebra: With a view toward algebraic geometry* which was published in 1995. Matthew Miller begins a detailed review with the following paragraph:-

With so many texts on commutative algebra available, one's reaction to this one might be "Why yet another one?", and "Why is it so fat?" The answer to the second question answers the first as well - this text has a distinctively different flavour than existing texts, both in coverage and style. Motivation and intuitive explanations appear throughout, there are many worked examples, and both text and problem sets lead up to contemporary research.

This important book earned Eisenbud the 2010 Steele prize from the American Mathematical Society. The citation indicates why the book is so significant:-

Eisenbud's book was designed with several purposes in mind. One was to provide an updated text on basic commutative algebra reflecting the intense activity in the field during the author's life. Another was to provide algebraic geometers, commutative algebraists, computational geometers, and other users of commutative algebra with a book where they could find results needed in their fields, especially those pertaining to algebraic geometry. But even more, Eisenbud felt that there was a great need for a book which did not present pure commutative algebra leaving the underlying geometry behind. In his introduction he writes, "It has seemed to me for a long time that commutative algebra is best practiced with knowledge of the geometric ideas that played a great role in its formation: in short, with a view toward algebraic geometry."

Two further books by Eisenbud are important. *The geometry of schemes* (2000), again written with Joe Harris:-

... is a wonderful introduction to the way of looking at algebraic geometry introduced by Alexandre Grothendieck and his school. The style of this book, however, differs greatly from that of Bourbaki; it is not formal and systematic, but friendly and inviting .... Thus this book introduces big ideas with seemingly simple, concrete examples, generalizes from them to an appropriate abstract formulation, and then applies the concept to interesting classical problems in a meaningful way. It is a pleasure to read.

Finally we mention Eisenbud's 2005 book *The geometry of syzygies. A second course in commutative algebra and algebraic geometry* and look forward to further fascinating books.

We have noted several honours given to Eisenbud, including the 2010 Steele Prize, but among those we have not yet mentioned is his election to the American Academy of Arts and Sciences in 2006 and the creation of the Eisenbud Professorship at Berkley's Mathematical Sciences Research Institute made possible by a US$10 million gift from the Simons Foundation in May 2007. We end this biography with his own descriptions of his non-mathematical interests:-

My interests outside mathematics include hiking, juggling, and, above all, music. Originally a flutist, I now spend most of my musical time singing art-songs(Schubert, Schumann, Brahms, Debussy, ...).

**Article by:** *J J O'Connor* and *E F Robertson*

**July 2011**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Eisenbud.html]