# Daniel Quillen

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Daniel_Quillen
> Markdown URL: https://mediated.wiki/source/Daniel_Quillen.md
> Source: https://en.wikipedia.org/wiki/Daniel_Quillen
> Source revision: 1354132773
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

American mathematician (1940–2011)

Daniel Quillen Born (1940-06-22)June 22, 1940 Orange, New Jersey, U.S. Died April 30, 2011(2011-04-30) (aged 70) Gainesville, Florida, U.S. Known for 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 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 American [mathematician](/source/Mathematician). He is known for being the "prime architect" of higher [algebraic *K*-theory](/source/Algebraic_K-theory), for which he was awarded the [Cole Prize](/source/Cole_Prize) in 1975 and the [Fields Medal](/source/Fields_Medal) in 1978.

From 1984 to 2006, he was the [Waynflete Professor of Pure Mathematics](/source/Waynflete_Professorships#Waynflete_Professors_of_Pure_Mathematics) at [Magdalen College, Oxford](/source/Magdalen_College%2C_Oxford).

## Education and career

Quillen was born in [Orange, New Jersey](/source/Orange%2C_New_Jersey), and attended [Newark Academy](/source/Newark_Academy). He entered [Harvard University](/source/Harvard_University), where he earned both his [AB](/source/Bachelor's_degree), in 1961, and his [PhD](/source/PhD) in 1964; the latter completed under the supervision of [Raoul Bott](/source/Raoul_Bott), with a thesis in [partial differential equations](/source/Partial_differential_equations). He was a [Putnam Fellow](/source/Putnam_Fellow) in 1959.[1]

Quillen obtained a position at the [Massachusetts Institute of Technology](/source/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](/source/Alfred_P._Sloan_Foundation) in Paris, during the academic year 1968–69, where he was greatly influenced by [Grothendieck](/source/Grothendieck), and then, during 1973–74, as a [Guggenheim Fellow](/source/Guggenheim_Fellow). In 1969–70, he was a visiting member of the [Institute for Advanced Study](/source/Institute_for_Advanced_Study) in [Princeton](/source/Princeton%2C_New_Jersey), where he came under the influence of [Sir Michael Atiyah](/source/Michael_Atiyah).

In 1978, Quillen received a [Fields Medal](/source/Fields_Medal) at the [International Congress of Mathematicians](/source/International_Congress_of_Mathematicians) held in [Helsinki](/source/Helsinki).[2]

From 1984 to 2006, he was the [Waynflete Professor of Pure Mathematics](/source/Waynflete_Professor_of_Pure_Mathematics) at [Magdalen College](/source/Magdalen_College), [Oxford](/source/Oxford).

Quillen retired at the end of 2006. He died from complications of [Alzheimer's disease](/source/Alzheimer's_disease) on April 30, 2011, aged 70.[3][4]

## 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 categories](/source/Model_category) and in particular the [Kan–Quillen model structure](/source/Kan%E2%80%93Quillen_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](/source/Adams_conjecture), formulated by [Frank Adams](/source/Frank_Adams), in [homotopy theory](/source/Homotopy_theory).[5] His proof of the conjecture used techniques from the [modular representation](/source/Modular_representation) theory of [groups](/source/Group_(mathematics)), which he later applied to work on [cohomology](/source/Cohomology) of groups and algebraic *K*-theory. He also worked on [complex cobordism](/source/Complex_cobordism), showing that its [formal group law](/source/Formal_group_law) is essentially the universal one.

In related work, he also supplied a proof of [Serre's conjecture](/source/Quillen%E2%80%93Suslin_theorem) about the triviality of algebraic [vector bundles](/source/Vector_bundle) on [affine space](/source/Affine_space), which led to the [Bass–Quillen conjecture](/source/Bass%E2%80%93Quillen_conjecture). He was also an architect (along with [Dennis Sullivan](/source/Dennis_Sullivan)) of [rational homotopy theory](/source/Rational_homotopy_theory).[6]

He introduced the [Quillen determinant line bundle](/source/Quillen_determinant_line_bundle) and the [Mathai–Quillen formalism](/source/Mathai%E2%80%93Quillen_formalism).

## See also

- [Friedhelm Waldhausen](/source/Friedhelm_Waldhausen)

## Selected publications

[Scholia](https://www.wikidata.org/wiki/Wikidata:Scholia) has a profile for [**Daniel Quillen (Q369658)**](https://iw.toolforge.org/scholia/Q369658).

- Quillen, Daniel G. ["Homology of commutative rings"](https://web.archive.org/web/20150420184538/http://chromotopy.org/paste/quillen.djvu). unpublished notes. Archived from [the original](http://chromotopy.org/paste/quillen.djvu) on 2015-04-20.

- Quillen, Daniel G. (1967). *Homotopical Algebra*. Lecture Notes in Mathematics. Vol. 43. Berlin, New York: [Springer-Verlag](/source/Springer-Verlag). [doi](/source/Doi_(identifier)):[10.1007/BFb0097438](https://doi.org/10.1007%2FBFb0097438). [ISBN](/source/ISBN_(identifier)) [978-3-540-03914-3](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-03914-3). [MR](/source/MR_(identifier)) [0223432](https://mathscinet.ams.org/mathscinet-getitem?mr=0223432).

- Quillen, Daniel (1969). ["On the formal group laws of unoriented and complex cobordism theory"](https://doi.org/10.1090%2FS0002-9904-1969-12401-8). *[Bulletin of the American Mathematical Society](/source/Bulletin_of_the_American_Mathematical_Society)*. **75** (6): 1293–1298. [doi](/source/Doi_(identifier)):[10.1090/S0002-9904-1969-12401-8](https://doi.org/10.1090%2FS0002-9904-1969-12401-8). [MR](/source/MR_(identifier)) [0253350](https://mathscinet.ams.org/mathscinet-getitem?mr=0253350).

- Quillen, D. (1969). "Rational homotopy theory". *Annals of Mathematics*. **90** (2): 205–295. [doi](/source/Doi_(identifier)):[10.2307/1970725](https://doi.org/10.2307%2F1970725). [JSTOR](/source/JSTOR_(identifier)) [1970725](https://www.jstor.org/stable/1970725). [MR](/source/MR_(identifier)) [0258031](https://mathscinet.ams.org/mathscinet-getitem?mr=0258031).

- Quillen, Daniel (1971). ["The Adams conjecture"](https://doi.org/10.1016%2F0040-9383%2871%2990018-8). *[Topology](/source/Topology_(journal))*. **10**: 67–80. [doi](/source/Doi_(identifier)):[10.1016/0040-9383(71)90018-8](https://doi.org/10.1016%2F0040-9383%2871%2990018-8). [ISSN](/source/ISSN_(identifier)) [0040-9383](https://search.worldcat.org/issn/0040-9383). [MR](/source/MR_(identifier)) [0279804](https://mathscinet.ams.org/mathscinet-getitem?mr=0279804).

- Quillen, Daniel (1971). "The spectrum of an equivariant cohomology ring. I". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. Second Series. **94** (3): 549–572. [doi](/source/Doi_(identifier)):[10.2307/1970770](https://doi.org/10.2307%2F1970770). [ISSN](/source/ISSN_(identifier)) [0003-486X](https://search.worldcat.org/issn/0003-486X). [JSTOR](/source/JSTOR_(identifier)) [1970770](https://www.jstor.org/stable/1970770). [MR](/source/MR_(identifier)) [0298694](https://mathscinet.ams.org/mathscinet-getitem?mr=0298694).

- Quillen, Daniel (1971). "The spectrum of an equivariant cohomology ring. II". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. Second Series. **94** (3): 573–602. [doi](/source/Doi_(identifier)):[10.2307/1970770](https://doi.org/10.2307%2F1970770). [ISSN](/source/ISSN_(identifier)) [0003-486X](https://search.worldcat.org/issn/0003-486X). [JSTOR](/source/JSTOR_(identifier)) [1970771](https://www.jstor.org/stable/1970771). [MR](/source/MR_(identifier)) [0298694](https://mathscinet.ams.org/mathscinet-getitem?mr=0298694).

- 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](/source/Springer-Verlag). pp. 85–147. [doi](/source/Doi_(identifier)):[10.1007/BFb0067053](https://doi.org/10.1007%2FBFb0067053). [ISBN](/source/ISBN_(identifier)) [978-3-540-06434-3](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-06434-3). [MR](/source/MR_(identifier)) [0338129](https://mathscinet.ams.org/mathscinet-getitem?mr=0338129).

- 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. [MR](/source/MR_(identifier)) [0422392](https://mathscinet.ams.org/mathscinet-getitem?mr=0422392). ([Quillen's Q-construction](/source/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](/source/Cambridge_University_Press). pp. 95–103. [MR](/source/MR_(identifier)) [0335604](https://mathscinet.ams.org/mathscinet-getitem?mr=0335604).

- Quillen, Daniel (1976). "Projective modules over polynomial rings". *[Inventiones Mathematicae](/source/Inventiones_Mathematicae)*. **36**: 167–171. [Bibcode](/source/Bibcode_(identifier)):[1976InMat..36..167Q](https://ui.adsabs.harvard.edu/abs/1976InMat..36..167Q). [doi](/source/Doi_(identifier)):[10.1007/BF01390008](https://doi.org/10.1007%2FBF01390008). [S2CID](/source/S2CID_(identifier)) [119678534](https://api.semanticscholar.org/CorpusID:119678534).

- Quillen, Daniel (1985). ["Superconnections and the Chern character"](https://doi.org/10.1016%2F0040-9383%2885%2990047-3). *[Topology](/source/Topology_(journal))*. **24** (1): 89–95. [doi](/source/Doi_(identifier)):[10.1016/0040-9383(85)90047-3](https://doi.org/10.1016%2F0040-9383%2885%2990047-3). [ISSN](/source/ISSN_(identifier)) [0040-9383](https://search.worldcat.org/issn/0040-9383). [MR](/source/MR_(identifier)) [0790678](https://mathscinet.ams.org/mathscinet-getitem?mr=0790678).

## References

1. **[^](#cite_ref-putnam_1-0)** ["The Mathematical Association of America's William Lowell Putnam Competition"](https://web.archive.org/web/20000229230334/http://www.maa.org/awards/putnam.html). Archived from [the original](http://www.maa.org/awards/putnam.html) on 2000-02-29. Retrieved 2013-03-28.

1. **[^](#cite_ref-2)** ["Home - International Mathematical Union (IMU)"](http://www.mathunion.org/index.php?id=prizewinners). *www.mathunion.org*.

1. **[^](#cite_ref-commalg_3-0)** ["commalg.org: Daniel Quillen"](http://www.commalg.org/2011/05/daniel-quillen). 2011. Retrieved 5 May 2011.

1. **[^](#cite_ref-4)** ["Daniel Quillen - Biography"](https://mathshistory.st-andrews.ac.uk/Biographies/Quillen/). *Maths History*. Retrieved 2025-04-02.

1. **[^](#cite_ref-segal_5-0)** Segal, Graeme (June 23, 2011), ["Daniel Quillen obituary"](https://www.theguardian.com/science/2011/jun/23/daniel-quillen-obituary), *[The Guardian](/source/The_Guardian)*

1. **[^](#cite_ref-6)** Quillen, D. (1969), "Rational homotopy theory", *[Annals of Mathematics](/source/Annals_of_Mathematics)*, **90** (2): 205–295, [doi](/source/Doi_(identifier)):[10.2307/1970725](https://doi.org/10.2307%2F1970725), [JSTOR](/source/JSTOR_(identifier)) [1970725](https://www.jstor.org/stable/1970725), [MR](/source/MR_(identifier)) [0258031](https://mathscinet.ams.org/mathscinet-getitem?mr=0258031)

## External links

- [Archive of Daniel Quillen’s notebooks for the years 1970 through 2003](http://www.claymath.org/publications/quillen-notebooks) at the [Clay Mathematics Institute](/source/Clay_Mathematics_Institute)

- O'Connor, John J.; [Robertson, Edmund F.](/source/Edmund_F._Robertson), ["Daniel Quillen"](https://mathshistory.st-andrews.ac.uk/Biographies/Quillen.html), *[MacTutor History of Mathematics Archive](/source/MacTutor_History_of_Mathematics_Archive)*, [University of St Andrews](/source/University_of_St_Andrews)

- [Daniel Quillen](https://mathgenealogy.org/id.php?id=13325) at the [Mathematics Genealogy Project](/source/Mathematics_Genealogy_Project)

- [Friedlander, Eric](/source/Eric_Friedlander); Grayson, Daniel (November 2012). ["Daniel Quillen"](https://www.ams.org/notices/201210/rtx121001392p.pdf) (PDF). *[Notices of the American Mathematical Society](/source/Notices_of_the_American_Mathematical_Society)*. **59** (10): 1392–1406. [doi](/source/Doi_(identifier)):[10.1090/noti903](https://doi.org/10.1090%2Fnoti903).

v t e Fields Medalists 1936 Ahlfors Douglas 1950 Schwartz Selberg 1954 Kodaira Serre 1958 Roth Thom 1962 Hörmander Milnor 1966 Atiyah Cohen Grothendieck Smale 1970 Baker Hironaka Novikov Thompson 1974 Bombieri Mumford 1978 Deligne Fefferman Margulis Quillen 1982 Connes Thurston Yau 1986 Donaldson Faltings Freedman 1990 Drinfeld Jones Mori Witten 1994 Bourgain Lions Yoccoz Zelmanov 1998 Borcherds Gowers Kontsevich McMullen 2002 Lafforgue Voevodsky 2006 Okounkov Perelman (declined) Tao Werner 2010 Lindenstrauss Ngô Smirnov Villani 2014 Avila Bhargava Hairer Mirzakhani 2018 Birkar Figalli Scholze Venkatesh 2022 Duminil-Copin Huh Maynard Viazovska Category Mathematics portal

Authority control databases International ISNI VIAF GND WorldCat National United States Czech Republic Netherlands Norway Poland Israel Belgium Academics CiNii Mathematics Genealogy Project zbMATH MathSciNet People DDB Other IdRef Yale LUX

---
Adapted from the Wikipedia article [Daniel Quillen](https://en.wikipedia.org/wiki/Daniel_Quillen) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Daniel_Quillen?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.