Frank Adams
Encyclopedia
John Frank Adams FRS (November 5, 1930 – January 7, 1989) was a British
mathematician
, one of the founders of homotopy theory.
, a suburb in south-east London
. He began research as a student of Abram Besicovitch
, but soon switched to algebraic topology
. He received his Ph.D. from the University of Cambridge
in 1956. His thesis, written under the direction of Shaun Wylie
, was titled On spectral sequences and self-obstruction invariants. He held the Fielden Chair
at the University of Manchester
(1964–1970), and became Lowndean Professor of Astronomy and Geometry
at the University of Cambridge (1970–1989). He was elected a Fellow of the Royal Society
in 1964.
His interests included mountaineering
—he would demonstrate how to climb right round a table at parties—and the game of Go.
He died in a car accident in Brampton, Cambridgeshire
. There is a memorial plaque for him in the Chapel of Trinity College, Cambridge
.
, but his innovations were always motivated by specific problems. Influenced by the French school of Henri Cartan
and Jean-Pierre Serre
, he reformulated and strengthened their method of killing homotopy groups
in spectral sequence
terms, creating the basic tool of stable homotopy theory
now known as the Adams spectral sequence
. This begins with Ext groups calculated over the ring of cohomology operation
s, which is the Steenrod algebra
in the classical case. He used this spectral sequence
to attack the celebrated Hopf invariant one problem, which he completely solved in a 1960 paper by making a deep analysis of secondary cohomology operation
s. The Adams–Novikov spectral sequence is an analogue of the Adams spectral sequence using an extraordinary cohomology theory in place of classical cohomology: it is a computational tool of great potential scope.
Adams was also a pioneer in the application of K-theory
. He invented the Adams operation
s in K-theory, which are derived from the exterior powers; they are now also widely used in purely algebraic contexts. Adams introduced them in a 1962 paper in order to solve the famous vector fields on spheres
problem. Subsequently he used them to investigate the Adams conjecture which is concerned (in one instance) with the image of the J-homomorphism
in the stable homotopy groups of spheres
. A later paper of Adams and Michael F. Atiyah uses the Adams operations to give an extremely elegant and much faster version of the above-mentioned Hopf invariant one result.
In 1974 Adams became the first recipient of the Senior Whitehead Prize
, awarded by the London Mathematical Society
.
Adams had many talented students, and was highly influential in the development of algebraic topology
in Britain and worldwide. His University of Chicago
lectures were published in a 1996 series titled "Chicago Lectures in Mathematics Series", such as Lectures On Exceptional Lie Groups and Stable Homotopy and Generalised Homology ISBN 0-226-00524-0 .
at the University of Manchester
is named in his honour.
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
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....
, one of the founders of homotopy theory.
Life
He was born in WoolwichWoolwich
Woolwich is a district in south London, England, located in the London Borough of Greenwich. The area is identified in the London Plan as one of 35 major centres in Greater London.Woolwich formed part of Kent until 1889 when the County of London was created...
, a suburb in south-east London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
. He began research as a student of Abram Besicovitch
Abram Samoilovitch Besicovitch
Abram Samoilovitch Besicovitch was a Russian mathematician, who worked mainly in England. He was born in Berdyansk on the Sea of Azov to a Karaite family.-Life and career:...
, but soon switched to algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
. He received his Ph.D. from the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
in 1956. His thesis, written under the direction of Shaun Wylie
Shaun Wylie
Shaun Wylie was a British mathematician and World War II codebreaker.-Early life:Wylie was born in Oxford, England, the fourth son of Sir Francis Wylie, later the first Warden of Rhodes House in Oxford. He was educated at Dragon School and then Winchester College...
, was titled On spectral sequences and self-obstruction invariants. He held the Fielden Chair
Fielden Chair of Pure Mathematics
The Fielden Chair of Pure Mathematics is an endowed professorial position in the School of Mathematics, University of Manchester, England.In 1870 Samuel Fielden, a wealthy mill owner from Todmorden, donated £150 to Owens College for the teaching of evening classes and a further £3000 for the...
at the University of Manchester
University of Manchester
The University of Manchester is a public research university located in Manchester, United Kingdom. It is a "red brick" university and a member of the Russell Group of research-intensive British universities and the N8 Group...
(1964–1970), and became Lowndean Professor of Astronomy and Geometry
Lowndean Professor of Astronomy and Geometry
The Lowndean chair of Astronomy and Geometry is one of the two major Professorships in Astronomy at Cambridge University, alongside the Plumian Professorship...
at the University of Cambridge (1970–1989). He was elected a Fellow of the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
in 1964.
His interests included mountaineering
Mountaineering
Mountaineering or mountain climbing is the sport, hobby or profession of hiking, skiing, and climbing mountains. While mountaineering began as attempts to reach the highest point of unclimbed mountains it has branched into specialisations that address different aspects of the mountain and consists...
—he would demonstrate how to climb right round a table at parties—and the game of Go.
He died in a car accident in Brampton, Cambridgeshire
Brampton, Cambridgeshire
Brampton – in Huntingdonshire , England – is a village near Godmanchester south west of Huntingdon. It has a population over 5000...
. There is a memorial plaque for him in the Chapel of Trinity College, Cambridge
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
.
Work
In the 1950s, homotopy theory was at an early stage of development, and unsolved problems abounded. Adams made a number of important theoretical advances in algebraic topologyAlgebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, but his innovations were always motivated by specific problems. Influenced by the French school of Henri Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
and Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, he reformulated and strengthened their method of killing homotopy groups
Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...
in spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
terms, creating the basic tool of stable homotopy theory
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
now known as the Adams spectral sequence
Adams spectral sequence
In mathematics, the Adams spectral sequence is a spectral sequence introduced by . Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory...
. This begins with Ext groups calculated over the ring of cohomology operation
Cohomology operation
In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from...
s, which is the Steenrod algebra
Steenrod algebra
In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.For a given prime number p, the Steenrod algebra Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p...
in the classical case. He used this spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
to attack the celebrated Hopf invariant one problem, which he completely solved in a 1960 paper by making a deep analysis of secondary cohomology operation
Cohomology operation
In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from...
s. The Adams–Novikov spectral sequence is an analogue of the Adams spectral sequence using an extraordinary cohomology theory in place of classical cohomology: it is a computational tool of great potential scope.
Adams was also a pioneer in the application of 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...
. He invented the Adams operation
Adams operation
In mathematics, an Adams operationis a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams...
s in K-theory, which are derived from the exterior powers; they are now also widely used in purely algebraic contexts. Adams introduced them in a 1962 paper in order to solve the famous vector fields on spheres
Vector fields on spheres
In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras....
problem. Subsequently he used them to investigate the Adams conjecture which is concerned (in one instance) with the image of the J-homomorphism
J-homomorphism
In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by , extending a construction of .-Definition:...
in the stable homotopy groups of spheres
Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...
. A later paper of Adams and Michael F. Atiyah uses the Adams operations to give an extremely elegant and much faster version of the above-mentioned Hopf invariant one result.
In 1974 Adams became the first recipient of the Senior Whitehead Prize
Senior Whitehead Prize
The Senior Whitehead Prize of the London Mathematical Society is currently awarded in odd numbered years in memory of John Henry Constantine Whitehead, president of the LMS between 1953 and 1955. The Prize is awarded to mathematicians normally resident in the United Kingdom on 1 January of the...
, awarded by the London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...
.
Adams had many talented students, and was highly influential in the development of algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
in Britain and worldwide. His University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
lectures were published in a 1996 series titled "Chicago Lectures in Mathematics Series", such as Lectures On Exceptional Lie Groups and Stable Homotopy and Generalised Homology ISBN 0-226-00524-0 .
Recognition
The main mathematics research seminar room in the Alan Turing BuildingAlan Turing Building
The Alan Turing Building, named after the mathematician and founder of computer science Alan Turing, is a building at the University of Manchester, in Manchester, England. It houses the School of Mathematics, the Photon Science Institute and the Jodrell Bank Centre for Astrophysics...
at the University of Manchester
University of Manchester
The University of Manchester is a public research university located in Manchester, United Kingdom. It is a "red brick" university and a member of the Russell Group of research-intensive British universities and the N8 Group...
is named in his honour.