John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
,
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
and low-dimensional holomorphic
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
. Milnor is a distinguished professor at
Stony Brook University
Stony Brook University (SBU), officially the State University of New York at Stony Brook, is a public research university in Stony Brook, New York. Along with the University at Buffalo, it is one of the State University of New York system' ...
and one of the five mathematicians to have won 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 h ...
, the
Wolf Prize
The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for ''"achievements in the interest of mankind and friendly relations among people ... irrespective of nati ...
, and the
Abel Prize
The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Pri ...
(the others being
Serre,
Thompson
Thompson may refer to:
People
* Thompson (surname)
* Thompson M. Scoon (1888–1953), New York politician
Places Australia
*Thompson Beach, South Australia, a locality
Bulgaria
* Thompson, Bulgaria, a village in Sofia Province
Canada ...
,
Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
, and
Margulis Margulis is a surname that, like its variants, is derived from the Ashkenazi Hebrew pronunciation of the Hebrew word ( Israeli Hebrew ), meaning 'pearl.' Notable people and characters with the name include:
* Berl Broder (born Margulis), Broder si ...
.)
Early life and career
Milnor was born on February 20, 1931, 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 ...
. His father was J. Willard Milnor and his mother was Emily Cox Milnor. As an undergraduate at
Princeton University
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 ...
he was named a
Putnam Fellow in 1949 and 1950
and also proved the
Fáry–Milnor theorem when he was only 19 years old. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of
Robert H. Fox. He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after completing a doctoral dissertation, titled "Isotopy of links", also under the supervision of Fox. His dissertation concerned
link groups (a generalization of the classical knot group) and their associated link structure, classifying
Brunnian links up to link-homotopy and introduced new invariants of it, called
Milnor invariants In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link com ...
. Upon completing his doctorate, he went on to work at Princeton. He was a professor at 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 schola ...
from 1970 to 1990.
He was an editor of the ''
Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as th ...
'' for a number of years after 1962. He has written a number of books which are famous for their clarity, presentation, and an inspiration for the research by many mathematicians in their areas even after many decades since their publication. He served as Vice President of the
AMS in 1976–77 period.
His students have included
Tadatoshi Akiba,
Jon Folkman
Jon Hal Folkman (December 8, 1938 – January 23, 1969) was an American mathematician, a student of John Milnor, and a researcher at the RAND Corporation.
Schooling
Folkman was a Putnam Fellow in 1960. He received his Ph.D. in 1964 from P ...
,
John Mather,
Laurent C. Siebenmann
Laurent Carl Siebenmann (the first name is sometimes spelled Laurence or Larry) (born 1939) is a Canadian mathematician based at the Université de Paris-Sud at Orsay, France.
After working for several years as a Professor at Orsay he became a ...
,
Michael Spivak
Michael David Spivak (25 May 19401 October 2020)Biographical sketch in Notices of the AMS', Vol. 32, 1985, p. 576. was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or- ...
, and Jonathan Sondow. His wife,
Dusa McDuff, is a professor of mathematics at
Barnard College
Barnard College of Columbia University is a private women's liberal arts college in the borough of Manhattan in New York City. It was founded in 1889 by a group of women led by young student activist Annie Nathan Meyer, who petitioned Columbia ...
and is known for her work in
symplectic topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Ha ...
.
Research
One of Milnor's best-known works is his proof in 1956 of the existence of
7-dimensional spheres
The Synchronized Position Hold Engage and Reorient Experimental Satellite (SPHERES) are a series of miniaturized satellites developed by MIT's Space Systems Laboratory for NASA and US Military, to be used as a low-risk, extensible test bed for the ...
with nonstandard differentiable structure, which marked the beginning of a new field – differential topology. He coined the term
exotic sphere
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of a ...
, referring to any ''n''-sphere with nonstandard differential structure. Kervaire and Milnor initiated the systematic study of exotic spheres, showing in particular that the 7-sphere has 15 distinct
differentiable structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for dif ...
s (28 if one considers orientation).
Egbert Brieskorn
Egbert Valentin Brieskorn (7 July 1936, in Rostock – 11 July 2013, in Bonn) was a German mathematician who introduced Brieskorn spheres and the Brieskorn–Grothendieck resolution.
Education
Brieskorn was born in 1936 as the son of a mill cons ...
found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a
singular point
Singularity or singular point may refer to:
Science, technology, and mathematics Mathematics
* Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
is diffeomorphic to these exotic spheres. Subsequently, Milnor worked on the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of isolated
singular points of complex hypersurfaces in general, developing the theory of the
Milnor fibration whose fiber has the
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
type of a bouquet of ''μ'' spheres where ''μ'' is known as the
Milnor number. Milnor's 1968 book on his theory, ''Singular Points of Complex Hypersurfaces'', inspired the growth of a huge and rich research area that continues to mature to this day.
In 1961 Milnor disproved the
Hauptvermutung
The ''Hauptvermutung'' of geometric topology is a now refuted conjecture asking whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the s ...
by illustrating two
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
es that are
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
but
combinatorially distinct, using the concept of
Reidemeister torsion
In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by and .
Analytic torsion (or Ray– ...
. This led to a wave of advances in topology by Milnor and many other mathematicians which changed the perception of the field forever.
In 1984 Milnor introduced a definition of
attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
. The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors.
Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarized by Peter Makienko in his review of ''Topological Methods in Modern Mathematics'':
It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems.
His other significant contributions include
microbundle In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964. It allows the creation of bundle-like objects in situations where they would not ordinarily be thought ...
s, influencing the usage of
Hopf algebra Hopf is a German surname. Notable people with the surname include:
* Eberhard Hopf (1902–1983), Austrian mathematician
* Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
* Heinz Hopf (actor) (1934–2001), Swe ...
s, theory of
quadratic forms and the related area of
symmetric bilinear forms, higher
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
,
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, and three-dimensional
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s.
Awards and honors
Milnor was elected as a member of the
American Academy of Arts and Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...
in 1961. In 1962 Milnor was awarded 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 h ...
for his work in differential topology. He was elected to the United States
National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nat ...
in 1963 and the
American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
1965. He later went on to win the
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social scienc ...
(1967), the
Lester R. Ford Award in 1970 and again in 1984, the
Leroy P. Steele Prize for "Seminal Contribution to Research" (1982), the
Wolf Prize
The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for ''"achievements in the interest of mankind and friendly relations among people ... irrespective of nati ...
in Mathematics (1989), the
Leroy P. Steele Prize for Mathematical Exposition (2004), and the
Leroy P. Steele Prize for Lifetime Achievement (2011). In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday.
Milnor was awarded the 2011
Abel Prize
The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Pri ...
, for his "pioneering discoveries in topology, geometry and algebra."
Reacting to the award, Milnor told the ''
New Scientist
''New Scientist'' is a magazine covering all aspects of science and technology. Based in London, it publishes weekly English-language editions in the United Kingdom, the United States and Australia. An editorially separate organisation publish ...
'' "It feels very good," adding that "
e is always surprised by a call at 6 o'clock in the morning."
In 2013 he became a
fellow
A fellow is a concept whose exact meaning depends on context.
In learned or professional societies, it refers to a privileged member who is specially elected in recognition of their work and achievements.
Within the context of higher education ...
of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...
, for "contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems".
In 2020 he received the
Lomonosov Gold Medal
The Lomonosov Gold Medal (russian: Большая золотая медаль имени М. В. Ломоносова ''Bol'shaya zolotaya medal' imeni M. V. Lomonosova''), named after Russian scientist and polymath Mikhail Lomonosov, is awarded ...
of the Russian Academy of Sciences.
Publications
Books
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*
*
*
*
*
*
*
Journal articles
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*
*
*
*
*
*
Lecture notes
*
See also
*
List of things named after John Milnor
*
Orbit portrait
In mathematics, an orbit portrait is a combinatorial tool used in complex dynamics for understanding the behavior of one-complex dimensional quadratic maps.
In simple words one can say that it is :
* a list of external angles for which rays lan ...
*
Microbundle In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964. It allows the creation of bundle-like objects in situations where they would not ordinarily be thought ...
References
External links
*
Home page at SUNYSBPhoto*
* (40 links from 1965 to May 2021, with 9 videos from Milnor's seminars)
{{DEFAULTSORT:Milnor, John
1931 births
20th-century American mathematicians
21st-century American mathematicians
Abel Prize laureates
Fields Medalists
Institute for Advanced Study faculty
Living people
Members of the United States National Academy of Sciences
Foreign Members of the Russian Academy of Sciences
National Medal of Science laureates
People from Orange, New Jersey
Princeton University alumni
Princeton University faculty
Putnam Fellows
Stony Brook University faculty
Topologists
Wolf Prize in Mathematics laureates
Fellows of the American Mathematical Society
Dynamical systems theorists
Geometers
Sloan Research Fellows
Members of the American Philosophical Society
Reeves family