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 Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...

Life

He was born in

Woolwich Woolwich () is a town in South London, 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 mainta ...

, a suburb in south-east London, and attended

Bedford School Bedford School is a 7–18 Single-sex education, boys Public school (United Kingdom), public school in the county town of Bedford in England. Founded in 1552, it is the oldest of four independent schools in Bedford run by the Harpur Trust. Bed ...

. He had a younger brother, Michael Adams, who rose to the rank of Air Vice-Marshal in the Royal Air Force. He began his academic career at

Trinity College, Cambridge Trinity College is a Colleges of the University of Cambridge, constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any ...

, as a student of Abram Besicovitch, 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 A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...

from the

University of Cambridge The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...

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 The University of Manchester is a public university, public research university in Manchester, England. The main campus is south of Manchester city centre, Manchester City Centre on Wilmslow Road, Oxford Road. The University of Manchester is c ...

(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 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, mountain climbing, or alpinism is a set of outdoor activities that involves ascending mountains. Mountaineering-related activities include traditional outdoor climbing, skiing, and traversing via ferratas that have become mounta ...

—he would demonstrate how to climb right round a table at parties (a Whitney traverse)—and the game of Go. He died in a car crash in

Brampton Brampton is a city in the Canadian Provinces and territories of Canada, province of Ontario, and the regional seat of the Regional Municipality of Peel. It is part of the Greater Toronto Area (GTA) and is a List of municipalities in Ontario#L ...

. There is a memorial plaque for him in the Chapel of

Work

In the 1950s,

was at an early stage of development, and unsolved problems abounded. Adams made a number of important theoretical advances in

, 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 c ...

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 inau ...

, he reformulated and strengthened their method of killing homotopy groups in

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 h ...

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 ...

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 c ...

. 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 stru ...

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 Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of ...

in the classical case. He used this

to attack the celebrated Hopf invariant one problem, which he completely solved in a 1960 paper by making a deep analysis of secondary cohomology operations. 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 c ...

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 geometr ...

. 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 introd ...

s in K-theory, which are derived from the

exterior power In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...

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 problem. Subsequently he used them to investigate the Adams conjecture, 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 de ...

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 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 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 r ...

, awarded by the

London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...

. He was a visiting scholar at the

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

in 1957–58.Institute for Advanced Study: A Community of Scholars
/ref> Adams had many talented students, and was highly influential in the development of

in Britain and worldwide. His

University of Chicago The University of Chicago (UChicago, Chicago, or UChi) is a Private university, private research university in Chicago, Illinois, United States. Its main campus is in the Hyde Park, Chicago, Hyde Park neighborhood on Chicago's South Side, Chic ...

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 at the

is named in his honour.

References

Publications

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External links