Daniel Gray "Dan" Quillen (born June 22, 1940) is an American mathematician and a Fields Medalist.
From 1984 to 2006 he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. He is renowned for being the "prime architect" of higher algebraic K-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978.
Quillen was born in Orange, New Jersey. He entered Harvard University, where he earned both his BA (1961) and his PhD (1964), the latter of which was completed under the supervision of Raoul Bott with a thesis in partial differential equations. He was a Putnam Fellow in 1959.
Quillen obtained a position at the Massachusetts Institute of Technology after completing his doctorate. However, he also spent a number of years at several other universities. This experience would prove to be important in influencing the direction of his research. He visited France twice: first as a Sloan Fellow in Paris, during the academic year 1968–69, where he was greatly influenced by Grothendieck, and then, during 1973–74, as a Guggenheim Fellow. In 1969–70, he was a visiting member of the Institute for Advanced Study in Princeton, where he came under the influence of Michael Atiyah.
In 1978, Quillen received a Fields Medal at the International Congress of Mathematicians held in Helsinki.
His Ph.D. students include Kenneth Brown, Howard Hiller, Jeanne Duflot, Mark Baker, Varghese Mathai (with whom he collaborated on the Mathai-Quillen formalism), and Jacek Brodzki.
Quillen retired at the end of 2006.
Quillen's most celebrated contribution (mentioned specifically in his Fields medal citation) was his formulation of higher algebraic K-theory in 1972, a problem that had baffled mathematicians since algebraic K-theory was first formulated. This new tool, formulated in terms of homotopy theory, proved to be successful in formulating and solving major problems in algebra, particularly in ring theory and module theory. More generally, Quillen developed tools (especially his theory of model categories) which allowed algebro-topological tools to be applied in other contexts.
Before his ground-breaking work in defining higher algebraic K-theory, Quillen worked on the Adams conjecture, formulated by Frank Adams in homotopy theory. His proof of the conjecture used techniques from the modular representation theory of groups, which he later applied to work on cohomology of groups and algebraic K-theory. He also worked on complex cobordism, showing that its formal group law is essentially the universal one.
In related work, he also supplied a proof of Serre's conjecture about the triviality of algebraic vector bundles on affine space.
He is also the architect (along with Dennis Sullivan) of rational homotopy theory.
* Quillen, D. (1969), "Rational homotopy theory", Annals of Math (Annals of Mathematics) 90 (2): 205–295, doi:10.2307/1970725, MR0258031, http://links.jstor.org/sici?sici=0003-486X%28196909%292%3A90%3A2%3C205%3ARHT%3E2.0.CO%3B2-S
* O'Connor, John J.; Robertson, Edmund F., "Daniel Quillen", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Quillen.html .