John Milnor
Encyclopedia
John Willard Milnor is an American
mathematician
known for his work in differential topology
, K-theory
and dynamical systems. He won the Fields Medal
in 1962, the Wolf Prize in 1989, and the Abel Prize
in 2011. Milnor is a distinguished professor at Stony Brook University. His wife, Dusa McDuff
, is a professor of mathematics at Barnard College
.
. As an undergraduate at Princeton University
he was named a Putnam Fellow
in 1949 and 1950 and also proved the Fary–Milnor theorem. He continued on to graduate school at Princeton and wrote his thesis, entitled "Isotopy of Link
s", which concerned link group
s (a generalization of the classical knot group) and their associated link structure. His advisor was Ralph Fox
. Upon completing his doctorate he went on to work at Princeton.
In 1962 Milnor was awarded the Fields Medal
for his work in differential topology. He later went on to win the National Medal of Science
(1967), the Leroy P Steele Prize for Seminal Contribution to Research (1982), the Wolf Prize in Mathematics (1989), the Leroy P Steele Prize for Mathematical Exposition (2004), and the Leroy P Steele Prize for Lifetime Achievement (2011). He was an editor of the Annals of Mathematics
for a number of years after 1962. He has written a number of books which are remarkable for their easy, clear and precise style.
Milnor was awarded the 2011 Abel Prize
, for his "pioneering discoveries in topology, geometry and algebra." Reacting to the award, Milnor told the New Scientist
"It feels very good,", adding that, "One is always surprised by a call at 6 o'clock in the morning."
His students have included Tadatoshi Akiba
, Jon Folkman
, John Mather, Laurent C. Siebenmann
, and Michael Spivak
.
spheres
with nonstandard differential structure. Later with Michel Kervaire
, he showed that the 7-sphere has 15 differentiable structures (28 if one considers orientation). An n-sphere with nonstandard differential structure is called an exotic sphere
, a term coined by Milnor. Egbert Brieskorn 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
is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points
of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fibre has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number
. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area which continues to develop to this day.
In 1961 Milnor disproved the Hauptvermutung
by exhibiting two simplicial complex
es which are homeomorphic
but combinatorially
distinct.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
known for his work in differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
and dynamical systems. He won the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1962, the Wolf Prize in 1989, and the Abel Prize
Abel Prize
The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...
in 2011. Milnor is a distinguished professor at Stony Brook University. His wife, Dusa McDuff
Dusa McDuff
Dusa McDuff is an English mathematician. She was born in London, England as the daughter of the noted biologist Conrad Hal Waddington. Her mother, Justin, born Justin Blanco White, was an architect, while her maternal grandmother was the feminist Amber Reeves, a lover of H.G. Wells and an author...
, is a professor of mathematics at Barnard College
Barnard College
Barnard College is a private women's liberal arts college and a member of the Seven Sisters. Founded in 1889, Barnard has been affiliated with Columbia University since 1900. The campus stretches along Broadway between 116th and 120th Streets in the Morningside Heights neighborhood in the borough...
.
Life
Milnor was born in Orange, New JerseyOrange, New Jersey
The City of Orange is a city and township in Essex County, New Jersey, United States. As of the 2010 United States Census, the township population was 30,134...
. As an undergraduate at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
he was named a Putnam Fellow
William Lowell Putnam Mathematical Competition
The William Lowell Putnam Mathematical Competition, often abbreviated to the Putnam Competition, is an annual mathematics competition for undergraduate college students of the United States and Canada, awarding scholarships and cash prizes ranging from $250 to $2,500 for the top students and $5,000...
in 1949 and 1950 and also proved the Fary–Milnor theorem. He continued on to graduate school at Princeton and wrote his thesis, entitled "Isotopy of Link
Link (knot theory)
In mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...
s", which concerned link group
Link group
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 Bachelor's thesis, .- Definition :...
s (a generalization of the classical knot group) and their associated link structure. His advisor was Ralph Fox
Ralph Fox
Ralph Hartzler Fox was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization and main-streaming of knot theory.Ralph Fox attended Swarthmore...
. Upon completing his doctorate he went on to work at Princeton.
In 1962 Milnor was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
for his work in differential topology. He later went on to win the National Medal of Science
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 sciences, biology, chemistry, engineering, mathematics and...
(1967), the Leroy P Steele Prize for Seminal Contribution to Research (1982), the Wolf Prize in Mathematics (1989), the Leroy P Steele Prize for Mathematical Exposition (2004), and the Leroy P Steele Prize for Lifetime Achievement (2011). He was an editor of the Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
for a number of years after 1962. He has written a number of books which are remarkable for their easy, clear and precise style.
Milnor was awarded the 2011 Abel Prize
Abel Prize
The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...
, for his "pioneering discoveries in topology, geometry and algebra." Reacting to the award, Milnor told the New Scientist
New Scientist
New Scientist is a weekly non-peer-reviewed English-language international science magazine, which since 1996 has also run a website, covering recent developments in science and technology for a general audience. Founded in 1956, it is published by Reed Business Information Ltd, a subsidiary of...
"It feels very good,", adding that, "One is always surprised by a call at 6 o'clock in the morning."
His students have included Tadatoshi Akiba
Tadatoshi Akiba
is a Japanese politician and served as the mayor of the city of Hiroshima, Japan from 1999 to 2011.- Early life :He studied mathematics at the University of Tokyo, receiving a B.S. in 1966 and an M.S. in 1968. He continued his studies under John Milnor at the Massachusetts Institute of Technology,...
, Jon Folkman
Jon Folkman
Jon Hal Folkman 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...
, John Mather, Laurent C. Siebenmann
Laurent C. Siebenmann
Laurent Carl Siebenmann 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 Directeur de Recherches in the French CNRS in 1976...
, and Michael Spivak
Michael Spivak
Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. He is the author of the five-volume Comprehensive Introduction to Differential Geometry. He received a Ph.D...
.
Work
His most celebrated single result is his proof in 1956 of the existence of 7-dimensionalSeven-dimensional space
In physics and mathematics, a sequence of n numbers can also be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional Euclidean space...
spheres
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
with nonstandard differential structure. Later with Michel Kervaire
Michel Kervaire
Michel André Kervaire was a French mathematician who made significant contributions to topology and algebra. He was the first to show the existence of topological n-manifolds with no differentiable structure , and computed the number of exotic spheres in dimensions greater than four...
, he showed that the 7-sphere has 15 differentiable structures (28 if one considers orientation). An n-sphere with nonstandard differential structure is called an exotic sphere
Exotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
, a term coined by Milnor. Egbert Brieskorn 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
Singular point of a curve
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.-Algebraic curves in the plane:...
is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points
Singular point of a curve
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.-Algebraic curves in the plane:...
of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fibre has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number
Milnor number
In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ.If f is a complex-valued holomorphic function germ then the Milnor number of f, denoted μ, is either an integer greater than or equal to zero, or it is infinite. It...
. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area which continues to develop to this day.
In 1961 Milnor disproved the Hauptvermutung
Hauptvermutung
The Hauptvermutung of geometric topology is the conjecture that any two triangulations of a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze.This conjecture is now known to be false...
by exhibiting two simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es which are homeomorphic
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
but combinatorially
Combinatorial topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces were regarded as derived from combinatorial decompositions such as simplicial complexes...
distinct.
See also
- Fary–Milnor theorem
- Milnor conjectureMilnor conjectureIn mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory of a general field F with characteristic different from 2, by means of the Galois cohomology of F with coefficients in Z/2Z. It was proved by .-Statement of the theorem:Let F be a field of...
in algebraic K-theoryAlgebraic K-theoryIn mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n.... - Milnor conjectureMilnor conjecture (topology)In knot theory, the Milnor conjecture says that the slice genus of the torus knot is/2.It is in a similar vein to the Thom conjecture....
in knot theoryKnot theoryIn topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a... - Milnor conjecture concerning manifolds with nonnegative Ricci curvatureRicci curvatureIn differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...
- Milnor fibration
- Milnor mapMilnor mapIn mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly...
- Milnor–Thurston kneading theoryMilnor–Thurston kneading theoryThe Milnor–Thurston kneading theory is a mathematical theory which analyzes the iterates of piecewise monotone mappings of an interval into itself...
- Orbit portraitOrbit portraitIn 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 :...