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36-452: John Frank Adams FRS (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory . He was born in Woolwich , a suburb in south-east London, and attended Bedford School . He began his academic career at Trinity College, Cambridge , as a student of Abram Besicovitch , but soon switched to algebraic topology . He received his PhD from

72-451: A sequence of abelian groups defined from a cochain complex . That is, cohomology is defined as the abstract study of cochains , cocycles , and coboundaries . Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology . Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in

108-519: A (finite) simplicial complex does have a finite presentation . Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are completely classified and are particularly easy to work with. In general, all constructions of algebraic topology are functorial ; the notions of category , functor and natural transformation originated here. Fundamental groups and homology and cohomology groups are not only invariants of

144-723: 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 . The main mathematics research seminar room in the Alan Turing Building at the University of Manchester is named in his honour. Fellow of the Royal Society Fellowship of the Royal Society ( FRS , ForMemRS and HonFRS )

180-679: A Chair (all of whom are Fellows of the Royal Society ). Members of the 10 Sectional Committees change every three years to mitigate in-group bias . Each Sectional Committee covers different specialist areas including: New Fellows are admitted to the Society at a formal admissions day ceremony held annually in July, when they sign the Charter Book and the Obligation which reads: "We who have hereunto subscribed, do hereby promise, that we will endeavour to promote

216-496: A combinatorial nature that allows for computation (often with a much smaller complex). An older name for the subject was combinatorial topology , implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW complex ). In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups , which led to

252-414: A convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces . The first and simplest homotopy group is the fundamental group , which records information about loops in a space. Intuitively, homotopy groups record information about

288-429: A great deal of manageable structure, often making these statements easier to prove. Two major ways in which this can be done are through fundamental groups , or more generally homotopy theory , and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with. The fundamental group of

324-548: A table at parties (a Whitney traverse)—and the game of Go . He died in a car crash in Brampton . There is a memorial plaque for him in the Chapel of Trinity College, Cambridge . 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 , but his innovations were always motivated by specific problems. Influenced by

360-1759: Is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural knowledge , including mathematics , engineering science , and medical science ". Fellowship of the Society, the oldest known scientific academy in continuous existence, is a significant honour. It has been awarded to many eminent scientists throughout history, including Isaac Newton (1672), Benjamin Franklin (1756), Charles Babbage (1816), Michael Faraday (1824), Charles Darwin (1839), Ernest Rutherford (1903), Srinivasa Ramanujan (1918), Jagadish Chandra Bose (1920), Albert Einstein (1921), Paul Dirac (1930), Winston Churchill (1941), Subrahmanyan Chandrasekhar (1944), Prasanta Chandra Mahalanobis (1945), Dorothy Hodgkin (1947), Alan Turing (1951), Lise Meitner (1955), Satyendra Nath Bose (1958), and Francis Crick (1959). More recently, fellowship has been awarded to Stephen Hawking (1974), David Attenborough (1983), Tim Hunt (1991), Elizabeth Blackburn (1992), Raghunath Mashelkar (1998), Tim Berners-Lee (2001), Venki Ramakrishnan (2003), Atta-ur-Rahman (2006), Andre Geim (2007), Bai Chunli (2014), James Dyson (2015), Ajay Kumar Sood (2015), Subhash Khot (2017), Elon Musk (2018), Elaine Fuchs (2019) and around 8,000 others in total, including over 280 Nobel Laureates since 1900. As of October 2018 , there are approximately 1,689 living Fellows, Foreign and Honorary Members, of whom 85 are Nobel Laureates. Fellowship of

396-725: Is confirmed by the Council in April, and a secret ballot of Fellows is held at a meeting in May. A candidate is elected if they secure two-thirds of votes of those Fellows voting. An indicative allocation of 18 Fellowships can be allocated to candidates from Physical Sciences and Biological Sciences; and up to 10 from Applied Sciences, Human Sciences and Joint Physical and Biological Sciences. A further maximum of six can be 'Honorary', 'General' or 'Royal' Fellows. Nominations for Fellowship are peer reviewed by Sectional Committees, each with at least 12 members and

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432-421: Is nominated by two Fellows of the Royal Society (a proposer and a seconder), who sign a certificate of proposal. Previously, nominations required at least five fellows to support each nomination by the proposer, which was criticised for supposedly establishing an old boy network and elitist gentlemen's club . The certificate of election (see for example ) includes a statement of the principal grounds on which

468-459: Is the study of mathematical knots . While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in three-dimensional Euclidean space , R 3 {\displaystyle \mathbb {R} ^{3}} . Two mathematical knots are equivalent if one can be transformed into

504-627: The Adams operations 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 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

540-594: 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

576-426: The post-nominal letters FRS . Every year, fellows elect up to ten new foreign members. Like fellows, foreign members are elected for life through peer review on the basis of excellence in science. As of 2016 , there are around 165 foreign members, who are entitled to use the post-nominal ForMemRS . Honorary Fellowship is an honorary academic title awarded to candidates who have given distinguished service to

612-792: 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 . He was a visiting scholar at the Institute for Advanced Study in 1957–58. 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

648-541: 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 operations , which is the Steenrod algebra in the classical case. He used this spectral sequence to attack

684-439: The Royal Society has been described by The Guardian as "the equivalent of a lifetime achievement Oscar " with several institutions celebrating their announcement each year. Up to 60 new Fellows (FRS), honorary (HonFRS) and foreign members (ForMemRS) are elected annually in late April or early May, from a pool of around 700 proposed candidates each year. New Fellows can only be nominated by existing Fellows for one of

720-655: The Society, we shall be free from this Obligation for the future". Since 2014, portraits of Fellows at the admissions ceremony have been published without copyright restrictions in Wikimedia Commons under a more permissive Creative Commons license which allows wider re-use. In addition to the main fellowships of the Royal Society (FRS, ForMemRS & HonFRS), other fellowships are available which are applied for by individuals, rather than through election. These fellowships are research grant awards and holders are known as Royal Society Research Fellows . In addition to

756-472: The award of Fellowship (FRS, HonFRS & ForMemRS) and the Research Fellowships described above, several other awards, lectures and medals of the Royal Society are also given. Algebraic topology Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for

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792-407: The basic shape, or holes, of a topological space. In algebraic topology and abstract algebra , homology (in part from Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group . In homology theory and algebraic topology, cohomology is a general term for

828-597: The cause of science, but do not have the kind of scientific achievements required of Fellows or Foreign Members. Honorary Fellows include the World Health Organization's Director-General Tedros Adhanom Ghebreyesus (2022), Bill Bryson (2013), Melvyn Bragg (2010), Robin Saxby (2015), David Sainsbury, Baron Sainsbury of Turville (2008), Onora O'Neill (2007), John Maddox (2000), Patrick Moore (2001) and Lisa Jardine (2015). Honorary Fellows are entitled to use

864-439: 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 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

900-443: The change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces. In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism (or more general homotopy ) of spaces. This allows one to recast statements about topological spaces into statements about groups, which have

936-515: The fellowships described below: Every year, up to 52 new fellows are elected from the United Kingdom, the rest of the Commonwealth of Nations , and Ireland, which make up around 90% of the society. Each candidate is considered on their merits and can be proposed from any sector of the scientific community. Fellows are elected for life on the basis of excellence in science and are entitled to use

972-619: The fundamental sense should assign "quantities" to the chains of homology theory. A manifold is a topological space that near each point resembles Euclidean space . Examples include the plane , the sphere , and the torus , which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be embedded in three dimensions, but can be embedded in four dimensions. Typically, results in algebraic topology focus on global, non-differentiable aspects of manifolds; for example Poincaré duality . Knot theory

1008-530: The good of the Royal Society of London for Improving Natural Knowledge, and to pursue the ends for which the same was founded; that we will carry out, as far as we are able, those actions requested of us in the name of the Council; and that we will observe the Statutes and Standing Orders of the said Society. Provided that, whensoever any of us shall signify to the President under our hands, that we desire to withdraw from

1044-439: The more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex . A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory . This class of spaces is broader and has some better categorical properties than simplicial complexes , but still retains

1080-552: The other via a deformation of R 3 {\displaystyle \mathbb {R} ^{3}} upon itself (known as an ambient isotopy ); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points , line segments , triangles , and their n -dimensional counterparts (see illustration). Simplicial complexes should not be confused with

1116-509: The post nominal letters HonFRS . Statute 12 is a legacy mechanism for electing members before official honorary membership existed in 1997. Fellows elected under statute 12 include David Attenborough (1983) and John Palmer, 4th Earl of Selborne (1991). The Council of the Royal Society can recommend members of the British royal family for election as Royal Fellow of the Royal Society . As of 2023 there are four royal fellows: Elizabeth II

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1152-546: The proposal is being made. There is no limit on the number of nominations made each year. In 2015, there were 654 candidates for election as Fellows and 106 candidates for Foreign Membership. The Council of the Royal Society oversees the selection process and appoints 10 subject area committees, known as Sectional Committees, to recommend the strongest candidates for election to the Fellowship. The final list of up to 52 Fellowship candidates and up to 10 Foreign Membership candidates

1188-513: The same Betti numbers as those derived through de Rham cohomology. This was extended in the 1950s, when Samuel Eilenberg and Norman Steenrod generalized this approach. They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms (e.g., a weak equivalence of spaces passes to an isomorphism of homology groups), verified that all existing (co)homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized

1224-449: The underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond—a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings. One of the first mathematicians to work with different types of cohomology

1260-449: Was Georges de Rham . One can use the differential structure of smooth manifolds via de Rham cohomology , or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question. De Rham showed that all of these approaches were interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were

1296-421: Was not a Royal Fellow, but provided her patronage to the society, as all reigning British monarchs have done since Charles II of England . Prince Philip, Duke of Edinburgh (1951) was elected under statute 12, not as a Royal Fellow. The election of new fellows is announced annually in May, after their nomination and a period of peer-reviewed selection. Each candidate for Fellowship or Foreign Membership

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