Daniel Quillen

1940–2011, Orange, New Jersey, USA, Havana, Florida, USA

AlgebraTopologyFields Medal 1978

American mathematician who founded higher algebraic K-theory and reshaped homotopy theory, rational homotopy theory and module theory through concepts such as the plus construction, model categories and the Quillen–Suslin theorem.

Daniel Gray Quillen was born on 22 June 1940 in Orange, New Jersey. He studied at Harvard University, earning his bachelor's degree in 1961 and his doctorate in 1964 under Raoul Bott; his thesis used partial differential equations to study overdetermined systems on manifolds. After positions at MIT he became a Sloan Fellow and spent formative visits at the IHÉS (with Grothendieck and Serre) and at Princeton, where Michael Atiyah strongly influenced his work.

Around 1970 Quillen produced the work for which he is best known: the definitions of higher algebraic K-groups via the plus construction and the Q-construction, the proof of their equivalence, and the foundations that turned K-theory into a systematic homotopical tool. In the same period he showed that the universal formal group law carries complex cobordism, he developed rational homotopy theory through differential graded Lie algebras, and in 1976 he proved Serre's conjecture on projective modules over polynomial rings — simultaneously with Suslin.

In 1984 Quillen moved to Oxford as Waynflete Professor of Pure Mathematics, a chair he held until retiring as emeritus in 2006. His late writings included work on cyclic homology and the theory of modules over finite groups. He died on 30 April 2011 in Florida, after several years of illness; Grothendieck's comment that Quillen was the student who most resembled him in style is often recalled in assessments of his legacy.

Workplaces: Massachusetts Institute of Technology, University of Oxford

United States

Award-winning work

Fields Medal 1978For the creation of higher algebraic K-theory. Quillen defined the higher K-groups of a ring through two constructions — the plus construction on the classifying space of the general linear group, and the Q-construction on categories — and proved they coincide, giving algebraic K-theory the structure of a homotopy theory with localization, devissage and resolution theorems. The same methods showed that formal group laws organize complex cobordism, explaining deep phenomena in stable homotopy theory. He also proved, independently of Andrei Suslin, the Serre conjecture that every finitely generated projective module over a polynomial ring over a field is free (the Quillen–Suslin theorem).

Contributions, linked to the library

Library topicHomological algebraHis higher algebraic K-theory and model categories are pillars of modern homological algebra.Library topicCommutative algebraThe Quillen–Suslin theorem: every finitely generated projective module over a polynomial ring over a field is free.