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Daniel Grey Quillen

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27 June 1940

Orange, New Jersey, USA

Dan Quillen's father trained as a chemical engineer but made his career as a physics teacher. Daniel attended Newark Academy and, from there, he entered Harvard University. He received his B.A. in 1961 and then began research at Harvard under R Bott's supervision. Quillen was awarded his Ph.D. for a thesis on partial differential equations in 1964 entitled Formal Properties of Over-Determined Systems of Linear Partial Differential Equations.

After receiving his doctorate, Quillen was appointed to the faculty of Massachusetts Institute of Technology. He spent a number of years undertaking research at other universities which were to prove important in setting the direction of his research. He was a Sloan Fellow in Paris during academic year 1968-69 when he was greatly influenced by Grothendieck , a visiting member of the Institute for Advanced Study at Princeton during 1969-70 when he was strongly influenced by Atiyah , and a Guggenheim Fellow again in France during 1973-74. Quillen at present works at the University of Oxford in England.

In the 1960s, Quillen described how to define the homology of simplical objects over many different categories, including sets, algebras over a ring, and unstable algebras over the Steenrod algebra.

Frank Adams had formulated a conjecture in homotopy theory which Quillen worked on. Quillen approached the Adams conjecture with two quite distinct approaches, namely using techniques from algebraic geometry and also using techniques from the modular representation theory of groups . Both approaches proved successful, the proof in the first approach being completed by one of Quillen's students, the second approach leading to a proof by Quillen.

The techniques using modular representation theory of groups were used by Quillen to great effect in later work on cohomology of groups and algebraic K-theory. The work on cohomology led to Quillen giving a structure theorem for mod p cohomology rings of finite groups, this structure theorem solving a number of open questions in the area.

Quillen received a Fields Medal at the International Congress of Mathematicians held in Helsinki in 1978. He received the award as the principal architect of the higher algebraic K-theory in 1972, a new tool that successfully used geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory.

Algebraic K-theory is an extension of ideas of Grothendieck to commutative rings. Grothendieck 's ideas were used by Atiyah and Hirzebruch when they created topological K-theory. Clearly Quillen's year spent in Paris under Grothendieck 's influence and at Princeton working with Atiyah were important factors in Quillen's development of algebraic K-theory.

Bass describes in how Quillen resolved the problem that the higher algebraic K-groups, Kn for n 3, being constructed in an essentially different way from the Grothendieck construction presented great difficulties:

... he borrowed techniques from homotopy theory, and in a completely novel way. The paper in which this so-called Q-construction occurs is essentially without mathematical precursors. Reading it for the first time is like landing on a new and friendly mathematical planet. One meets there not only new theorems and new methods, but new mathematical creatures and a complete paradigm of gestures for dealing with them. Higher algebraic K-theory is effectively built there from first principles and, in 63 pages, reaches a state of maturity that one normally expects from the efforts of several mathematicians over several years.

As to his character, this is shown in :

When Quillen received his Ph.D. at the age of 24, he and his wife Jean, a violinist, were already caring for two of their five children. His precocity as a mathematician and as a father perhaps influenced the early greying of his hair, but it has not altered his boyish look or his easy and modest manner. He has a somewhat retiring life-style, appearing rarely in public, and then almost invariably with some extraordinary new theorem or idea in hand.

In Hyman Bass sums up Quillen's contribution leading up to the award of the Fields Medal in 1978 as follows:

Mathematical talent tends to express itself either in problem solving or in theory building. It is with rare cases like Quillen that one has the satisfaction of seeing hard, concrete problems solved with general ideas of great force and scope and by the unification of methods from diverse fields of mathematics. Quillen has had a deep impact on the perceptions and the very thinking habits of a whole generation of young algebraists and topologists. One studies his work not only to be informed, but to be edified.

Source:School of Mathematics and Statistics University of St Andrews, Scotland