Daniel Quillen
Daniel Quillen  

Born 
Orange, New Jersey 
June 22, 1940
Died  April 30, 2011 Haven Hospice,^{[1]} North Florida 
(aged 70)
Nationality  American 
Known for  Algebraic Ktheory (Quillen's Qconstruction), Quillen–Suslin theorem, Bass–Quillen conjecture, rational homotopy theory, Quillen determinant line bundle, Mathai–Quillen formalism, Quillen's lemma, Quillen metric, Quillen's theorems A and B 
Awards 
Fields Medal (1978) Cole Prize (1975) Putnam Fellow (1959) 
Scientific career  
Fields  Mathematics 
Thesis  Formal Properties of OverDetermined Systems of Linear Partial Differential Equations (1964) 
Doctoral advisor  Raoul Bott 
Doctoral students  Kenneth Brown 
Influences 
Alexander Grothendieck JeanPierre Serre 
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician.
From 1984 to 2006, he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. He is known for being the "prime architect" of higher algebraic Ktheory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978.
Contents
Education and career
Quillen was born in Orange, New Jersey, and attended Newark Academy. He entered Harvard University, where he earned both his AB, in 1961, and his PhD in 1964; the latter completed under the supervision of Raoul Bott, with a thesis in partial differential equations. He was a Putnam Fellow in 1959.^{[2]}
Quillen obtained a position at the Massachusetts Institute of Technology after completing his doctorate. He also spent a number of years at several other universities. 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.^{[3]}
Quillen retired at the end of 2006. He died from complications of Alzheimer's disease on April 30, 2011, aged 70, in Florida.^{[4]}
Mathematical contributions
Quillen's best known contribution (mentioned specifically in his Fields medal citation) was his formulation of higher algebraic Ktheory in 1972. This new tool, formulated in terms of homotopy theory, proved to be successful in formulating and solving problems in algebra, particularly in ring theory and module theory. More generally, Quillen developed tools (especially his theory of model categories) which allowed algebrotopological tools to be applied in other contexts.
Before his work in defining higher algebraic Ktheory, Quillen worked on the Adams conjecture, formulated by Frank Adams, in homotopy theory.^{[5]} 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 Ktheory. 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, which led to the Bass–Quillen conjecture. He was also an architect (along with Dennis Sullivan) of rational homotopy theory.^{[6]}
He introduced the Quillen determinant line bundle and the Mathai–Quillen formalism.
Selected publications
 Quillen, Daniel G., Homology of commutative rings, unpublished notes, archived from the original on 20150420
 Quillen, Daniel G. (1967), Homotopical algebra, Lecture Notes in Mathematics, 43, Berlin, New York: SpringerVerlag, doi:10.1007/BFb0097438, ISBN 9783540039143, MR 0223432
 Quillen, Daniel (1969), "On the formal group laws of unoriented and complex cobordism theory", Bulletin of the American Mathematical Society, 75: 1293–1298, doi:10.1090/S000299041969124018, MR 0253350
 Quillen, D. (1969), "Rational homotopy theory", Annals of Math, Annals of Mathematics, 90 (2): 205–295, doi:10.2307/1970725, JSTOR 1970725, MR 0258031
 Quillen, Daniel (1971), "The Adams conjecture", Topology. An International Journal of Mathematics, 10: 67–80, doi:10.1016/00409383(71)900188, ISSN 00409383, MR 0279804
 Quillen, Daniel (1971), "The spectrum of an equivariant cohomology ring. I", Annals of Mathematics, Second Series, 94: 549–572, ISSN 0003486X, JSTOR 1970770, MR 0298694
 Quillen, Daniel (1971), "The spectrum of an equivariant cohomology ring. II", Annals of Mathematics, Second Series, 94: 573–602, ISSN 0003486X, JSTOR 1970771, MR 0298694
 Quillen, Daniel (1973), "Higher algebraic Ktheory. I", Algebraic Ktheory, I: Higher Ktheories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math, 341, Berlin, New York: SpringerVerlag, pp. 85–147, doi:10.1007/BFb0067053, MR 0338129
 Quillen, Daniel (1975), "Higher algebraic Ktheory", Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Montreal, Quebec: Canad. Math. Congress, pp. 171–176, MR 0422392 (Quillen's Qconstruction)
 Quillen, Daniel (1974), "Higher Ktheory for categories with exact sequences", New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), London Math. Soc. Lecture Note Ser., 11, Cambridge University Press, pp. 95–103, MR 0335604
 Quillen, Daniel (1976), "Projective modules over polynomial rings", Inventiones Mathematicae, 36: 167–171, doi:10.1007/BF01390008
 Quillen, Daniel (1985), "Superconnections and the Chern character", Topology. An International Journal of Mathematics, 24 (1): 89–95, doi:10.1016/00409383(85)900473, ISSN 00409383, MR 0790678
References
 ^ Daniel Quillen (entry posted on Sunday, May 1, 2011)

^ "The Mathematical Association of America's William Lowell Putnam Competition". Retrieved 20130328.
first2=
missinglast2=
in Authors list (help)  ^ http://www.mathunion.org/index.php?id=prizewinners

^ "commalg.org: Daniel Quillen". 2011. Retrieved 05032011. Check date values in:
accessdate=
(help)  ^ Segal, Graeme (June 23, 2011), "Daniel Quillen obituary", The Guardian
 ^ Quillen, D. (1969), "Rational homotopy theory", Annals of Math, 90 (2): 205–295, doi:10.2307/1970725, JSTOR 1970725, MR 0258031
External links
 Archive of Daniel Quillen’s notebooks for the years 1970 through 2003 at the Clay Mathematics Institute
 O'Connor, John J.; Robertson, Edmund F., "Daniel Quillen", MacTutor History of Mathematics archive, University of St Andrews.
 Daniel Quillen at the Mathematics Genealogy Project
 Friedlander, Eric; Grayson, Daniel (November 2012), "Daniel Quillen" (PDF), Notices of the American Mathematical Society, Providence, RI: American Mathematical Society, 59 (10): 1392–1406, doi:10.1090/noti903
 1940 births
 2011 deaths
 20thcentury mathematicians
 21stcentury mathematicians
 Deaths from Alzheimer's disease
 Members of the United States National Academy of Sciences
 Fields Medalists
 Putnam Fellows
 Topologists
 Institute for Advanced Study visiting scholars
 Harvard University alumni
 People from Orange, New Jersey
 ISI highly cited researchers
 Waynflete Professors of Pure Mathematics
 Fellows of Magdalen College, Oxford
 Diseaserelated deaths in Florida
 Guggenheim Fellows
 Mathematicians from New Jersey