Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the

Cole Prize The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number ...

in 1975 and the

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...

in 1978. From 1984 to 2006, he was the

Waynflete Professor of Pure Mathematics Waynflete could refer to: *William Waynflete (1395–1486), English Lord Chancellor and bishop of Winchester *Waynflete Professorships at Magdalen College, Oxford, named after William Waynflete *Waynflete School, a private day school in Portland, M ...

Magdalen College, Oxford Magdalen College (, ) is a constituent college of the University of Oxford. It was founded in 1458 by William of Waynflete. Today, it is the fourth wealthiest college, with a financial endowment of £332.1 million as of 2019 and one of the s ...

Education and career

Quillen was born in

Orange, New Jersey The City of Orange is a township in Essex County, in the U.S. state of New Jersey. As of the 2010 U.S. census, the township's population was 30,134, reflecting a decline of 2,734 (−8.3%) from the 32,868 counted in 2000. Orange was original ...

, and attended

Newark Academy Newark Academy is a coeducational private day school located in Livingston, in Essex County, New Jersey, United States, serving students in sixth through twelfth grades. The school has been accredited by the Middle States Association of Col ...

. He entered

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...

, where he earned both his AB, in 1961, and his PhD in 1964; the latter completed under the supervision of

Raoul Bott Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions whi ...

, with a thesis in

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

. He was a

Putnam Fellow The William Lowell Putnam Mathematical Competition, often abbreviated to Putnam Competition, is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the United States and Canada (regar ...

in 1959. Quillen obtained a position at the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...

after completing his doctorate. He also spent a number of years at several other universities. He visited France twice: first as a

Sloan Fellow The Sloan Fellows program is the world's first mid-career and senior career master's degree in general management and leadership. It was initially supported by a grant from Alfred P. Sloan, the late CEO of General Motors, to his alma mater, MI ...

in Paris, during the academic year 1968–69, where he was greatly influenced by Grothendieck, and then, during 1973–74, as a

Guggenheim Fellow Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...

. In 1969–70, he was a visiting member of the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

Princeton Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ni ...

, where he came under the influence of

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...

. In 1978, Quillen received a

at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...

held in

Helsinki Helsinki ( or ; ; sv, Helsingfors, ) is the Capital city, capital, primate city, primate, and List of cities and towns in Finland, most populous city of Finland. Located on the shore of the Gulf of Finland, it is the seat of the region of U ...

. From 1984 to 2006, he was the

Magdalen College Magdalen College (, ) is a constituent college of the University of Oxford. It was founded in 1458 by William of Waynflete. Today, it is the fourth wealthiest college, with a financial endowment of £332.1 million as of 2019 and one of the s ...

Oxford Oxford () is a city in England. It is the county town and only city of Oxfordshire. In 2020, its population was estimated at 151,584. It is north-west of London, south-east of Birmingham and north-east of Bristol. The city is home to the ...

. Quillen retired at the end of 2006. He died from complications of

Alzheimer's disease Alzheimer's disease (AD) is a neurodegeneration, neurodegenerative disease that usually starts slowly and progressively worsens. It is the cause of 60–70% of cases of dementia. The most common early symptom is difficulty in short-term me ...

on 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 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) 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 John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory. Life He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began researc ...

, in homotopy theory. His proof of the conjecture used techniques from the

modular representation Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as h ...

theory of

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

, which he later applied to work on

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

of groups and algebraic ''K''-theory. He also worked on

complex cobordism In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it ...

, showing that its

formal group law In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one o ...

is essentially the universal one. In related work, he also supplied a proof of Serre's conjecture about the triviality of algebraic

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...

s on

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...

, which led to the Bass–Quillen conjecture. He was also an architect (along with

Dennis Sullivan Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Ce ...

) of

rational homotopy theory In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homo ...

. He introduced the

Quillen determinant line bundle In mathematics, the Quillen determinant line bundle is a line bundle over the space of Cauchy–Riemann operators of a vector bundle over a Riemann surface, introduced by . Quillen proved the existence of the Quillen metric on the determinant line ...

and the Mathai–Quillen formalism.

Selected publications

* * * * * * * * * ( Quillen's Q-construction) * * *

References

External links

Archive of Daniel Quillen’s notebooks for the years 1970 through 2003
at the Clay Mathematics Institute * * * {{DEFAULTSORT:Quillen, Daniel 1940 births 2011 deaths 20th-century American mathematicians 21st-century American mathematicians Deaths from dementia in Florida Deaths from Alzheimer's disease Members of the United States National Academy of Sciences Fields Medalists Putnam Fellows Algebraists Topologists Institute for Advanced Study visiting scholars Harvard University alumni People from Orange, New Jersey Waynflete Professors of Pure Mathematics Fellows of Magdalen College, Oxford Mathematicians from New Jersey Massachusetts Institute of Technology School of Science faculty Newark Academy alumni

Education and career

Mathematical contributions

See also

Selected publications

References

External links