{{short description|American mathematician (1940–2011)}} {{Infobox scientist | name = Daniel Quillen | image = DQuillen.jpg | birth_date = {{Birth date|1940|6|22}} | birth_place = [[Orange, New Jersey]], U.S. | death_date = {{Death date and age|2011|4|30|1940|6|22}} | death_place = [[Gainesville, Florida]], U.S. | ethnicity = | field = [[Mathematics]] | work_institutions = | alma_mater = | doctoral_advisor = [[Raoul Bott]] | doctoral_students = [[Kenneth Brown (mathematician)|Kenneth Brown]]<br>[[Varghese Mathai]] | thesis_title = Formal Properties of Over-Determined Systems of Linear Partial Differential Equations | thesis_year = 1964 | thesis_url = https://search.worldcat.org/title/76994794 | known_for = [[Algebraic K-theory|Algebraic ''K''-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]], [[Kan–Quillen model structure]] | prizes = [[Fields Medal]] (1978)<br>[[Cole Prize]] (1975)<br>[[Putnam Fellow]] (1959) }}
'''Daniel Gray Quillen''' (June 22, 1940 – April 30, 2011) was an American [[mathematician]]. He is known for being the "prime architect" of higher [[algebraic K-theory|algebraic ''K''-theory]], for which he was awarded the [[Cole Prize]] in 1975 and the [[Fields Medal]] in 1978.
From 1984 to 2006, he was the [[Waynflete Professorships#Waynflete Professors of Pure Mathematics|Waynflete Professor of Pure Mathematics]] at [[Magdalen College, Oxford]].
==Education and career== Quillen was born in [[Orange, New Jersey]], and attended [[Newark Academy]]. He entered [[Harvard University]], where he earned both his [[bachelor's degree|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.<ref name=putnam>{{cite web| title=The Mathematical Association of America's William Lowell Putnam Competition| url=http://www.maa.org/awards/putnam.html| access-date=2013-03-28| archive-date=2000-02-29| archive-url=https://web.archive.org/web/20000229230334/http://www.maa.org/awards/putnam.html| url-status=dead}}</ref>
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 [[Alfred P. Sloan Foundation|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, New Jersey|Princeton]], where he came under the influence of [[Michael Atiyah|Sir Michael Atiyah]].
In 1978, Quillen received a [[Fields Medal]] at the [[International Congress of Mathematicians]] held in [[Helsinki]].<ref>{{cite web|url=http://www.mathunion.org/index.php?id=prizewinners|title=Home - International Mathematical Union (IMU)|website=www.mathunion.org}}</ref>
From 1984 to 2006, he was the [[Waynflete Professor of Pure Mathematics]] at [[Magdalen College]], [[Oxford]].
Quillen retired at the end of 2006. He died from complications of [[Alzheimer's disease]] on April 30, 2011, aged 70.<ref name=commalg>{{cite web|title=commalg.org: Daniel Quillen|url=http://www.commalg.org/2011/05/daniel-quillen|year=2011|access-date= 5 May 2011}}</ref><ref>{{Cite web |title=Daniel Quillen - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Quillen/ |access-date=2025-04-02 |website=Maths History |language=en}}</ref>
==Mathematical contributions== Quillen's best known contribution (mentioned specifically in his Fields medal citation) was his formulation of higher algebraic ''K''-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 of [[Model category|model categories]] and in particular the [[Kan–Quillen model structure]]) that allowed algebro-topological tools to be applied in other contexts.
Before his work in defining higher algebraic ''K''-theory, Quillen worked on the [[Adams conjecture]], formulated by [[Frank Adams]], in [[homotopy theory]].<ref name=segal>{{citation |last=Segal |first=Graeme |title=Daniel Quillen obituary | newspaper=[[The Guardian]] |date=June 23, 2011 |url=https://www.theguardian.com/science/2011/jun/23/daniel-quillen-obituary}}</ref> His proof of the conjecture used techniques from the [[modular representation]] theory of [[Group (mathematics)|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 [[Quillen–Suslin theorem|Serre's conjecture]] about the triviality of algebraic [[vector bundle]]s on [[affine space]], which led to the [[Bass–Quillen conjecture]]. He was also an architect (along with [[Dennis Sullivan]]) of [[rational homotopy theory]].<ref>{{citation|mr=0258031|first=D. |last=Quillen|title=Rational homotopy theory|journal=[[Annals of Mathematics]]|volume=90|year=1969|pages=205–295|doi=10.2307/1970725|jstor=1970725|issue=2}}</ref>
He introduced the [[Quillen determinant line bundle]] and the [[Mathai–Quillen formalism]].
==See also== * [[Friedhelm Waldhausen]]
==Selected publications== {{Scholia}} * {{cite web|last1=Quillen |first1=Daniel G. |title=Homology of commutative rings |url=http://chromotopy.org/paste/quillen.djvu |series=unpublished notes |url-status=dead |archive-url=https://web.archive.org/web/20150420184538/http://chromotopy.org/paste/quillen.djvu |archive-date=2015-04-20 }} * {{Cite book | last1=Quillen | first1=Daniel G. | title=Homotopical Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-03914-3 | doi=10.1007/BFb0097438 | mr=0223432 | year=1967 | volume=43}} * {{cite journal |mr=0253350| author-link=Daniel Quillen| last= Quillen|first= Daniel |title=On the formal group laws of unoriented and complex cobordism theory|journal= [[Bulletin of the American Mathematical Society]] |volume=75 | issue=6|year=1969 |pages=1293–1298 |doi=10.1090/S0002-9904-1969-12401-8 |doi-access=free }} * {{cite journal|mr=0258031|first=D. |last=Quillen|title=Rational homotopy theory|journal=Annals of Mathematics|volume=90|year=1969|pages=205–295|doi=10.2307/1970725|jstor=1970725|issue=2}} * {{Cite journal | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=The Adams conjecture | doi=10.1016/0040-9383(71)90018-8 | mr=0279804 | year=1971 | journal=[[Topology (journal)|Topology]] | issn=0040-9383 | volume=10 | pages=67–80| doi-access=free }} * {{Cite journal | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=The spectrum of an equivariant cohomology ring. I | jstor=1970770 | mr=0298694 | year=1971 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=94 | issue=3 | pages=549–572 | doi=10.2307/1970770}} * {{Cite journal | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=The spectrum of an equivariant cohomology ring. II | jstor=1970771 | mr=0298694 | year=1971 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=94 | issue=3 | pages=573–602 | doi=10.2307/1970770}} * {{Cite book | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math | doi=10.1007/BFb0067053 | mr=0338129 | year=1973 | volume=341 | chapter=Higher algebraic K-theory. I | pages=85–147| isbn=978-3-540-06434-3 }} * {{Cite book | last1=Quillen | first1=Daniel | author1-link= Daniel Quillen | title=Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1 | publisher=Canad. Math. Congress | location=Montreal, Quebec | mr=0422392 | year=1975 | chapter=Higher algebraic K-theory | pages=171–176}} ([[Quillen's Q-construction]]) * {{Cite book | last1=Quillen | first1=Daniel | title=New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972) | publisher=[[Cambridge University Press]] | series=London Math. Soc. Lecture Note Ser. | mr=0335604 | year=1974 | volume=11 | chapter=Higher K-theory for categories with exact sequences | pages=95–103}} * {{Cite journal | last = Quillen | first = Daniel | author-link = Daniel Quillen | year = 1976 | title = Projective modules over polynomial rings | journal = [[Inventiones Mathematicae]] | volume = 36 | pages = 167–171 | doi = 10.1007/BF01390008 | bibcode = 1976InMat..36..167Q | s2cid = 119678534 }} * {{Cite journal | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=Superconnections and the Chern character | doi=10.1016/0040-9383(85)90047-3 | mr=790678 | year=1985 | journal=[[Topology (journal)|Topology]] | issn=0040-9383 | volume=24 | issue=1 | pages=89–95| doi-access=free }}
== References == {{reflist}}
== External links == * [http://www.claymath.org/publications/quillen-notebooks Archive of Daniel Quillen’s notebooks for the years 1970 through 2003] at the [[Clay Mathematics Institute]] * {{MacTutor Biography|id=Quillen}} * {{MathGenealogy |id=13325}} * {{Cite journal | last1 = Friedlander | first1 = Eric |author-link=Eric Friedlander | last2 = Grayson | first2 = Daniel | title = Daniel Quillen | journal = [[Notices of the American Mathematical Society]] | volume = 59 | issue = 10 | pages = 1392–1406 | date = November 2012 | url = https://www.ams.org/notices/201210/rtx121001392p.pdf | doi=10.1090/noti903| doi-access = free }}
{{Fields medalists}} {{Authority control}}
{{DEFAULTSORT:Quillen, Daniel}} [[Category:1940 births]] [[Category:2011 deaths]] [[Category:20th-century American mathematicians]] [[Category:21st-century American mathematicians]] [[Category:Deaths from Alzheimer's disease in Florida]] [[Category:Members of the United States National Academy of Sciences]] [[Category:Fields Medalists]] [[Category:Putnam Fellows]] [[Category:American algebraists]] [[Category:American topologists]] [[Category:Institute for Advanced Study visiting scholars]] [[Category:Harvard University alumni]] [[Category:People from Orange, New Jersey]] [[Category:Waynflete Professors of Pure Mathematics]] [[Category:Fellows of Magdalen College, Oxford]] [[Category:Mathematicians from New Jersey]] [[Category:MIT School of Science faculty]] [[Category:Newark Academy alumni]]