John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
.
Life
He was born in
Woolwich
Woolwich () is a district in southeast London, England, within the Royal Borough of Greenwich.
The district's location on the River Thames led to its status as an important naval, military and industrial area; a role that was maintained throu ...
, a suburb in south-east London, and attended
Bedford School
:''Bedford School is not to be confused with Bedford Girls' School, Bedford High School, Bedford Modern School, Old Bedford School in Bedford, Texas or Bedford Academy in Bedford, Nova Scotia.''
Bedford School is a public school (English indep ...
. He began research as a student of
Abram Besicovitch
Abram Samoilovitch Besicovitch (or Besikovitch) (russian: link=no, Абра́м Само́йлович Безико́вич; 23 January 1891 – 2 November 1970) was a Russian mathematician, who worked mainly in England. He was born in Berdyansk ...
, but soon switched to
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. He received his
PhD PHD or PhD may refer to:
* Doctor of Philosophy (PhD), an academic qualification
Entertainment
* '' PhD: Phantasy Degree'', a Korean comic series
* ''Piled Higher and Deeper'', a web comic
* Ph.D. (band), a 1980s British group
** Ph.D. (Ph.D. albu ...
from the
University of Cambridge
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...
in 1956. His thesis, written under the direction of
University of Manchester
, mottoeng = Knowledge, Wisdom, Humanity
, established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Univer ...
(1964–1970), and became
Lowndean Professor of Astronomy and Geometry
The Lowndean chair of Astronomy and Geometry is one of the two major Professorships in Astronomy (alongside the Plumian Professorship) and a major Professorship in Mathematics at Cambridge University. It was founded in 1749 by Thomas Lowndes, an ...
at the University of Cambridge (1970–1989). He was elected a Fellow of the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
in 1964.
His interests included
mountaineering
Mountaineering or alpinism, is a set of outdoor activities that involves ascending tall mountains. Mountaineering-related activities include traditional outdoor climbing, skiing, and traversing via ferratas. Indoor climbing, sport climbing, a ...
—he would demonstrate how to climb right round a table at parties (a
Whitney Whitney may refer to:
Film and television
* ''Whitney'' (2015 film), a Whitney Houston biopic starring Yaya DaCosta
* ''Whitney'' (2018 film), a documentary about Whitney Houston
* ''Whitney'' (TV series), an American sitcom that premiered i ...
traverse)—and the game of Go.
He died in a car crash in
Brampton
Brampton ( or ) is a city in the Canadian Provinces and territories of Canada, province of Ontario. Brampton is a city in the Greater Toronto Area (GTA) and is a List of municipalities in Ontario#Lower-tier municipalities, lower-tier municipalit ...
. There is a memorial plaque for him in the Chapel of
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Founded in 1546 by Henry VIII, King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge ...
.
Work
In the 1950s,
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
was at an early stage of development, and unsolved problems abounded. Adams made a number of important theoretical advances in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, but his innovations were always motivated by specific problems. Influenced by the French school of
Henri Cartan
Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology.
He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of co ...
and
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...
terms, creating the basic tool of
stable homotopy theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the F ...
now known as the
Adams spectral sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now ca ...
. This begins with
Ext group
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic st ...
s calculated over the ring of
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 cohomo ...
s, which is the
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 A_p is the graded Hopf algebra over the field \mathbb_p of order p, c ...
in the classical case. He used this
spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...
to attack the celebrated
Hopf invariant one
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
__TOC__ Motivation
In 1931 Heinz Hopf used Clifford parallels to construct the ''Hopf map''
:\eta\colon S^3 \to S^ ...
problem, which he completely solved in a 1960 paper by making a deep analysis of
secondary cohomology operation In mathematics, a secondary cohomology operation is a functorial correspondence between cohomology groups. More precisely, it is a natural transformation from the kernel of some primary cohomology operation to the cokernel of another primary operati ...
s. The
Adams–Novikov spectral sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now ca ...
is an analogue of the Adams spectral sequence using an
extraordinary cohomology theory
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 viewed ...
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
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
. He invented the
Adams operation In mathematics, an Adams operation, denoted ψ''k'' for natural numbers ''k'', is 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 introduce ...
s in K-theory, which are derived from the
exterior power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
s; they are now also widely used in purely algebraic contexts. Adams introduced them in a 1962 paper to solve the famous
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.
Specifically, the question is how many l ...
problem. Subsequently he used them to investigate the
Adams conjecture
Adams may refer to:
* For persons, see Adams (surname)
Places United States
*Adams, California
*Adams, California, former name of Corte Madera, California
*Adams, Decatur County, Indiana
*Adams, Kentucky
*Adams, Massachusetts, a New England town ...
, which is concerned (in one instance) with the image of the
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
Whitehead's original homomorphism is d ...
in the stable
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 ...
. A later paper of Adams and
Michael F. 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 ...
uses the Adams operations to give an extremely elegant and much faster version of the above-mentioned
Hopf invariant one
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
__TOC__ Motivation
In 1931 Heinz Hopf used Clifford parallels to construct the ''Hopf map''
:\eta\colon S^3 \to S^ ...
result.
In 1974 Adams became the first recipient of the
Senior Whitehead Prize
The Senior Whitehead Prize of the London Mathematical Society (LMS) is now 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 ...
, awarded by the
London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...
. He was a visiting scholar 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 scholar ...
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
in Britain and worldwide. His
University of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...
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'' .
Recognition
The main mathematics research seminar room in the
Alan 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 th ...
at the
University of Manchester
, mottoeng = Knowledge, Wisdom, Humanity
, established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Univer ...
is named in his honour.
See also
*
Adams filtration
In mathematics, especially in the area of algebraic topology known as stable homotopy theory, the Adams filtration and the Adams–Novikov filtration allow a stable homotopy group to be understood as built from layers, the ''n''th layer containin ...