**Hyman Bass**'s parents were immigrants to the United States from Lithuania. They had a large family with Hyman being the seventh of his parents eight children. It consisted of four girls, including Madeleine and Sylvia, all older than Hyman, and four boys, Leon and Manuel being older than Hyman, with Isaac the youngest in the family. Hyman Bass writes [1]:-

I absorbed little family cultural influence during my childhood, apart from my father's orthodox Judaism, and the heated family discussions of national and world politics.

Hyman was only seven years old when World War II began, and nine years old when the United States entered the war after Japanese aircraft attacked the US fleet at Pearl Harbour on 7 December 1941. His older brothers and sisters were old enough to serve in the war and this was to have a major influence on Hyman. In particular his brother Manuel took part in the V12 officers training programme. He [1]:-

... came home on leave speaking excitedly to me and my younger brother Isaac about the science courses he was taking. It was these conversations that first awakened my awareness of and interest in science.

After World War II ended, Leon and Manuel ended their war service and began studying at the California Institute of Technology. The family, including Hyman, moved to Los Angeles at this time. When the time was approaching for Bass to begin university studies he asked his brother Manuel for advice. In particular, although he was very interested by science, he wanted to study at a liberal arts college and Manuel suggested that Princeton would be the best place. Bass won a scholarship to enable him to attend, then entered Princeton thinking that he would probably major in physics.

One of the courses in which Bass enrolled in his first year of study was a calculus course lectured by Emil Artin and tutored by Serge Lang and John Tate. It was to completely change the direction of Bass's interests [1]:-

This was a rigorous course in analysis, with everything proved. More than that, Artin's lectures, which were like pieces of theatre, made you understand the crux of a proof, the moments where some essential new idea or invention makes its appearance. ... I had no reason to believe that this course was substantially different from any other college calculus course. I later learned otherwise, not only because of its remarkable teaching staff, but also for the students then enrolled. Our class included several students who later became prominent mathematicians...

Bass graduated from Princeton with a B.A. in 1955 and then went to the University of Chicago as an NSF graduate fellow to study for his Masters Degree. Although he found Chicago very different from Princeton, nevertheless it was "again an environment that lived and breathed mathematics." [1]:-

Kaplansky gave an inspiring series of courses on homological methods in commutative algebra. Faithful to his expository style, Kaplansky aimed not to lead us deeply into the whole forest, but rather to show us a few magnificent trees. He would target a few central theorems, and then track a logical geodesic to them, using the greatest economy of means, eschewing any avoidable extrinsic machinery. These lectures, attended by both students and young faculty, initiated a generation of Chicago students, myself included, into this new and blossoming field.

Inspired by Irving Kaplansky, after the award of an M.S. in 1956, Bass undertook work for his doctorate under his supervision. He published his first mathematics paper *Finite monadic algebras* in 1958. The idea of a "monadic algebra" had been introduced by Paul Halmos, who was on the faculty in Chicago at this time, in a paper in 1956. Halmos defined such algebras as Boolean algebras with a distributive closure operation, in which the complement of any closed element is closed. Bass gave explicit descriptions of the finite monadic algebras in his first paper, showing they were in one-one correspondence to finite sets of positive integers.

He examined refinements of the notions of homological dimension in his doctoral thesis *Global Dimensions of Rings* and he received his Ph.D. from the University of Chicago for this thesis in August 1959. The natural route for Bass to take at this stage would have been to take up a fellowship at the Institute for Advanced Study and concentrate solely on research. However Samuel Eilenberg was at Columbia University in Morningside Heights, Manhattan, New York and he persuaded Bass to take a teaching job there. Bass writes in [1]:-

Since great teachers had been much of my mathematical inspiration, I looked forward to teaching, so it was not a hard sell.

Appointed as Ritt Instructor at Columbia [1]:-

... there was no one directly engaged with the kind of mathematics I had been thinking about. So I tried to learn what others were doing, attending many of the graduate courses: number theory and algebraic geometry from Lang, Lie groups and class field theory from Harish-Chandra, differential algebra from Kolchin, category theory from Eilenberg, and fiber bundles from Albrecht Dold.

Bass produced a series of papers during his first years at Columbia, for example *Finitistic dimension and a homological generalization of semiprimary rings* (1960), *Projective modules over algebras* (1961), *Injective dimension in noetherian rings* (1962), and *Torsion free and projective modules* (1962). The paper *The homotopy theory of projective modules* (1962) was written jointly with S Schanuel. The authors write:-

Serre has established the rudiments of a dictionary for translating the language of projective modules into that of vector bundles. With this point of departure we have attempted to adapt some of the results and methods of homotopy theory to certain purely arithmetic and even noncommutative settings. Detailed proofs of our results will appear elsewhere.

Bass spent the year 1963-64 as an NSF Postdoctoral Fellow at the College de France. An outcome of this year was the paper *Sous-groupes d'indice fini dans* *SL*(*n*, **Z**) (published in 1964) written jointly with J-P Serre and M Lazard. Returning to Columbia University, he was promoted to Assistant Professor in 1963.

Columbia University came into existence in 1912 but Columbia College had existed for well over 100 years before that. Columbia College only educated men (up to 1983), but it was affiliated to Barnard College for women from its foundation in 1889. Bass served as Associate Professor and Chair at Barnard during 1964-65. From 1964 to 1966 he was also a Sloan Fellow. He was promoted to Professor of Mathematics at Columbia in 1965, a position he held until 1992 when he was named Adrain Professor. During his period as professor, Bass was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques in Paris during the year 1968-69, then served as Chair of the Department of Mathematics at Columbia University from 1975 to 1979.

We have mentioned some areas of Bass's work above, but let us note that he himself gives his research interests as algebraic K-theory; number theory; group theory (geometric methods); and algebraic geometry. To many people, Bass is best known for his classic text *Algebraic K-theory* published in 1968. A Heller, in a review of the book, writes:-

Algebraic K-theory flows from two sources. The(J H C)Whiteheadtorsion, introduced in order to study the topological notion of simple homotopy type, leads to the groups K_{1}. The Grothendieck group of projective modules over a ring leads to K_{0}. The latter notion was applied by Atiyah and Hirzebruch in order to construct a new cohomology theory which has been enormously fruitful in topology. The observation that these two ideas could be unified in a beautiful and powerful theory with widespread applications in algebra is due to the author, who is also responsible for a major portion of those applications. The author of this book is thus uniquely qualified to produce a fundamental document in this new and expanding field; he has also produced an admirable exposition of its methods and results.

For this outstanding book Bass was awarded the Van Amringe Prize from Columbia University in the year following its publication. In [2] Bass recounts his contributions in this area. He writes:-

These informal reminiscences, presented at the ICTP2002Conference on algebraic K-theory, recount the trajectory in the author's early research, from work on the Serre conjecture(on projective modules over polynomial algebras), via ideas from algebraic geometry and topology, to the ideas and constructions that eventually contributed to the founding of algebraic K-theory. The solution of the congruence subgroup problem is presented as a pivotal event.

He has received many other honours and prizes, in addition to that for *Algebraic K-theory*, such as the Cole Prize in Algebra from the American Mathematical Society in 1975. He was elected to the American Academy of Arts and Sciences in 1980, the National Academy of Sciences in 1982, and the American Association for the Advancement of Science in 1986 serving as Chairman of Section A during 1987-1988 and 1997-1998.

Bass has visited many leading institutions around the world, spending varying periods of time. For example he was a visiting professor at the Universidad Nacional Autonoma de Mexico in the summer of 1965. He was first a visiting professor at the Tata Institute of Fundamental Research in Bombay, India, in the winter of 1965-66. There he gave the course *Topics in Algebraic K-theory* which was published in* Lecture notes, Tata Institute of Fundamental Research* (Bombay, 1966). Further visits to the Tata Institute took place in the summer of 1969, the summer of 1976, the autumn of 1979, the autumn of 1990, and the summer of 1995. Other examples of such visits are Trinity College, University of Cambridge, England for a term in 1973, the Instituto de Matematica Pura e Applicada, Rio de Janeiro, in the summer of 1977, the University of Utah, Salt Lake City, in the autumn in 1977, then the University of California, Berkeley, in the winter of 1977-78. Over the years he has, as one would expect, spent several periods as a visiting member of the Institute for Advanced Study in Princeton; in the summer of 1964, the academic year 1965-66, the summer of 1975, and the summer of 1979.

Although Bass was interested in mathematical education throughout his career, this interest took a different turn in 1991 when he was invited to join the Mathematical Sciences Education Board at the National Academy of Sciences. His description of the relation between mathematicians and educators given in [1] is well worth quoting:-

Mathematicians tend to think of educational matters almost exclusively in terms of content; what material should be taught. They thus approach teachers and educators in the guise of experts with the answers in hand, ready to contribute their authorative advice. They often convey disdain(even if unintended)to the teachers and educators with whom they speak, and inspire defensiveness and resentment in return. One result of this history is that the important conversations that now need to take place between mathematicians and educators are burdened with suspicion and cultural prejudices.

He has written many papers on mathematical education such as *Education reform from a national perspective: the mathematics community's investment and future* (1994), *Mathematicians as educators* (1997), *Mathematicians and the national eighth grade test* (1997), and *Mathematicians and the National Eighth-Grade Test* (1998). All these articles were published in the *Notices of the American Mathematical Society*. He has also published many articles jointly with Deborah L Ball, an educational researcher and elementary school teacher. Examples of such articles are *Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics* (2000), *Making believe: The collective construction of public mathematical knowledge in the elementary classroom* (2000), *Making mathematics reasonable in school* (2003), *A practice-based theory of mathematical knowledge for teaching: The case of mathematical reasoning* (2003), and *Knowing mathematics for teaching* (2003).

Finally we mention that Bass has served the American Mathematical Society in many roles. For example he served as Vice President in 1980-81, Chair of the Committee on Education from 1995 to 2000, and President from 2001 to 2003. We end this article by quoting from [4] Bass's views on the biggest challenges facing the mathematics profession:-

There are two perennial issues. One has to do with resources to support the research enterprise. That is a constant campaign with federal and public agencies. And the other is whether we are drawing enough talent into the field to maintain quality and productivity.

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