Daniel Quillen

Daniel Quillen
Born	(1940-06-22)June 22, 1940 Orange, New Jersey
Died	April 30, 2011(2011-04-30)(aged 70) Haven Hospice,North Florida
Nationality	American
Known for	AlgebraicK-theory(Quillen's Q-construction),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 Over-Determined Systems of Linear Partial Differential Equations(1964)
Doctoral advisor	Raoul Bott
Doctoral students	Kenneth Brown Varghese Mathai

Daniel Gray Quillen(June 22, 1940 – April 30, 2011) was an Americanmathematician.He is known for being the "prime architect" of higheralgebraicK-theory,for which he was awarded theCole Prizein 1975 and theFields Medalin 1978.

From 1984 to 2006, he was theWaynflete Professor of Pure MathematicsatMagdalen College, Oxford.

Education and career

Quillen was born inOrange, New Jersey,and attendedNewark Academy.He enteredHarvard University,where he earned both hisAB,in 1961, and hisPhDin 1964; the latter completed under the supervision ofRaoul Bott,with a thesis inpartial differential equations.He was aPutnam Fellowin 1959.

Quillen obtained a position at theMassachusetts Institute of Technologyafter completing his doctorate. He also spent a number of years at several other universities. He visited France twice: first as aSloan Fellowin Paris, during the academic year 1968–69, where he was greatly influenced byGrothendieck,and then, during 1973–74, as aGuggenheim Fellow.In 1969–70, he was a visiting member of theInstitute for Advanced StudyinPrinceton,where he came under the influence ofMichael Atiyah.

In 1978, Quillen received aFields Medalat theInternational Congress of Mathematiciansheld inHelsinki.

From 1984 to 2006, he was theWaynflete Professor of Pure MathematicsatMagdalen College,Oxford.

Quillen retired at the end of 2006. He died from complications ofAlzheimer's diseaseon April 30, 2011, aged 70, in Florida.

Mathematical contributions

Quillen's best known contribution (mentioned specifically in his Fields medal citation) was his formulation of higher algebraicK-theory 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 ofmodel categories) that allowed algebro-topological tools to be applied in other contexts.

Before his work in defining higher algebraicK-theory, Quillen worked on theAdams conjecture,formulated byFrank Adams,inhomotopy theory.His proof of the conjecture used techniques from themodular representationtheory ofgroups,which he later applied to work oncohomologyof groups and algebraicK-theory. He also worked oncomplex cobordism,showing that itsformal group lawis essentially the universal one.

In related work, he also supplied a proof ofSerre's conjectureabout the triviality of algebraicvector bundlesonaffine space,which led to theBass–Quillen conjecture.He was also an architect (along withDennis Sullivan) ofrational homotopy theory.

He introduced theQuillen determinant line bundleand theMathai–Quillen formalism.

See also

• Friedhelm Waldhausen

Selected publications

• Quillen, Daniel G."Homology of commutative rings".unpublished notes. Archived fromthe originalon 2015-04-20.
• Quillen, Daniel G. (1967).Homotopical Algebra.Lecture Notes in Mathematics. Vol. 43. Berlin, New York:Springer-Verlag.doi:10.1007/BFb0097438.ISBN978-3-540-03914-3.MR0223432.
• Quillen, Daniel(1969)."On the formal group laws of unoriented and complex cobordism theory".Bulletin of the American Mathematical Society.75(6): 1293–1298.doi:10.1090/S0002-9904-1969-12401-8.MR0253350.
• Quillen, D. (1969). "Rational homotopy theory".Annals of Mathematics.90(2): 205–295.doi:10.2307/1970725.JSTOR1970725.MR0258031.
• Quillen, Daniel(1971)."The Adams conjecture".Topology.10:67–80.doi:10.1016/0040-9383(71)90018-8.ISSN0040-9383.MR0279804.
• Quillen, Daniel(1971). "The spectrum of an equivariant cohomology ring. I".Annals of Mathematics.Second Series.94(3): 549–572.doi:10.2307/1970770.ISSN0003-486X.JSTOR1970770.MR0298694.
• Quillen, Daniel(1971). "The spectrum of an equivariant cohomology ring. II".Annals of Mathematics.Second Series.94(3): 573–602.doi:10.2307/1970770.ISSN0003-486X.JSTOR1970771.MR0298694.
• Quillen, Daniel(1973). "Higher algebraic K-theory. I".Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972).Lecture Notes in Math. Vol. 341. Berlin, New York:Springer-Verlag.pp. 85–147.doi:10.1007/BFb0067053.ISBN978-3-540-06434-3.MR0338129.
• Quillen, Daniel(1975). "Higher algebraic K-theory".Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1.Montreal, Quebec: Canad. Math. Congress. pp. 171–176.MR0422392.(Quillen's Q-construction)
• Quillen, Daniel (1974). "Higher K-theory for categories with exact sequences".New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972).London Math. Soc. Lecture Note Ser. Vol. 11.Cambridge University Press.pp. 95–103.MR0335604.
• Quillen, Daniel(1976). "Projective modules over polynomial rings".Inventiones Mathematicae.36:167–171.Bibcode:1976InMat..36..167Q.doi:10.1007/BF01390008.S2CID119678534.
• Quillen, Daniel(1985)."Superconnections and the Chern character".Topology.24(1): 89–95.doi:10.1016/0040-9383(85)90047-3.ISSN0040-9383.MR0790678.

External links

• Archive of Daniel Quillen’s notebooks for the years 1970 through 2003at theClay Mathematics Institute
• O'Connor, John J.;Robertson, Edmund F.,"Daniel Quillen",MacTutor History of Mathematics Archive,University of St Andrews
• Daniel Quillenat theMathematics Genealogy Project
• Friedlander, Eric; Grayson, Daniel (November 2012)."Daniel Quillen"(PDF).Notices of the American Mathematical Society.59(10): 1392–1406.doi:10.1090/noti903.