Frank Adams
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John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory.


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 thr ...
, a suburb in south-east London, and attended Bedford School. 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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. He received his PhD from the
University of Cambridge The University of Cambridge is a public collegiate research university in Cambridge, England. Founded in 1209 and granted a royal charter by Henry III in 1231, Cambridge is the world's third oldest surviving university and one of its most pr ...
in 1956. His thesis, written under the direction of
Shaun Wylie Shaun Wylie (17 January 1913 – 2 October 2009Fielden Chair 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 Univ ...
(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—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 province of Ontario. Brampton is a city in the Greater Toronto Area (GTA) and is a lower-tier municipality within Peel Region. The city has a population of 656,480 as of the 2021 Census, making it ...
. 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 King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge or Oxford. ...
.


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 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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, but his innovations were always motivated by specific problems. Influenced by the French school of Henri Cartan 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 ...
, 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 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 groups 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 coho ...
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 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 viewe ...
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 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 ...
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 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 o ...
. 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 ...
, awarded by the London Mathematical Society. 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 schola ...
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
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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
in Britain and worldwide. His
University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of 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
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 Univ ...
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 containing ...


References


Publications

* *


External links

* {{DEFAULTSORT:Adams, Frank 1930 births 1989 deaths Adams, John Frank Adams, John Frank Adams, John Frank Adams, John Frank Academics of the Victoria University of Manchester Academics of the University of Cambridge Adams, John Frank Adams, John Frank Adams, John Frank Adams, John Frank Fellows of the Royal Society Foreign associates of the National Academy of Sciences Lowndean Professors of Astronomy and Geometry Institute for Advanced Study visiting scholars Homotopy theory People from Woolwich