Daniel Grey Quillen

Born: 27 June 1940 in Orange, New Jersey, USA
Died: 30 April 2011 in Gainsville, Florida, USA

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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 [2] 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 [2]:-

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 [they now have six]. 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 [2] 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.

From 1984 to 2006 Quillen was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. In 2000 the journal K-Theory issued a special part dedicated to Quillen on the occasion of his sixtieth birthday. In the same year he gave the Erdős Colloquium at the University of Florida on Module theory for nonunital rings. On 22 May 2006 the 39th K-theory Day at Oxford was set up to celebrate Quillen's 65th birthday. Jacek Brodzki lectured on Analysis and geometry on discrete groups, Mathai Varghese lectured on T-duality and non-commutative geometry, Joachim Cuntz lectured on K-theory for locally convex algebras, and Eric Friedlander closed the proceedings with the lecture Dan and me: looking back at some of Dan's remarkable mathematics. Having reached the age of 65, Quillen retired in 2006.

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

Click on this link to see a list of the Glossary entries for this page

List of References (5 books/articles)

A Poster of Dan Quillen

Mathematicians born in the same country

Additional Material in MacTutor

  1. Obituary: Guardian

Honours awarded to Dan Quillen
(Click below for those honoured in this way)
Speaker at International Congress1974
AMS Cole Prize for algebra1975
Fields Medal1978
BMC plenary speaker1983
Waynflete Chair1984 - 2006

Cross-references in MacTutor

  1. Chronology: 1970 to 1980

Other Web sites
  1. Encyclopaedia Britannica
  2. Astroseti (A Spanish translation of this biography)
  1. S Lansberg
  2. Mathematical Genealogy Project

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