J. W. S. Cassels
Encyclopedia
John William Scott Cassels (also known as "Ian Cassels"), FRS (born 11 July 1922 in Durham
) is a leading English
mathematician
.
and George Heriot's School
in Edinburgh
, Cassels graduated from the University of Edinburgh
with an MA
in 1943.
His academic career was interrupted in World War II
when he was involved in cryptography
at Bletchley Park
. After the war he became a research student of Louis Mordell
at Trinity College
, Cambridge
; he received his PhD
in 1949 and was elected a fellow of Trinity in the same year.
After finishing his major, Cassels spent a year lecturing in mathematics at the University of Manchester
before returning to Cambridge as a lecturer in 1950. He was appointed Reader
in Arithmetic in 1963, the same year he was elected as a fellow of the Royal Society of London. In 1967 he was appointed as Sadleirian Professor of Pure Mathematics at Cambridge. In 1969 he became Head of the Department of Pure Mathematics and Mathematical Statistics
. He retired in 1984.
s. After a period when he worked on geometry of numbers
and diophantine approximation
, he returned in the later 1950s to the arithmetic of elliptic curves, writing a series of papers connecting the Selmer group with Galois cohomology
and laying some of the foundations of the modern theory of infinite descent
. His best-known single result may be the proof that the Tate-Shafarevich group, if it is finite, must have order that is a square; the proof being by construction of an alternating form.
Cassels has often studied individual Diophantine equation
s by algebraic number theory
and p-adic methods.
His publications include 200 papers. His advanced textbooks have influenced generations of mathematicians; some of Cassels's books have remained in print for decades.
Durham
Durham is a city in north east England. It is within the County Durham local government district, and is the county town of the larger ceremonial county...
) is a leading English
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
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....
.
Biography
Educated at Neville's Cross Council School in DurhamDurham
Durham is a city in north east England. It is within the County Durham local government district, and is the county town of the larger ceremonial county...
and George Heriot's School
George Heriot's School
George Heriot's School is an independent primary and secondary school on Lauriston Place in the Old Town of Edinburgh, Scotland, with around 1600 pupils, 155 teaching staff and 80 non-teaching staff. It was established in 1628 as George Heriot's Hospital, by bequest of the royal goldsmith George...
in Edinburgh
Edinburgh
Edinburgh is the capital city of Scotland, the second largest city in Scotland, and the eighth most populous in the United Kingdom. The City of Edinburgh Council governs one of Scotland's 32 local government council areas. The council area includes urban Edinburgh and a rural area...
, Cassels graduated from the University of Edinburgh
University of Edinburgh
The University of Edinburgh, founded in 1583, is a public research university located in Edinburgh, the capital of Scotland, and a UNESCO World Heritage Site. The university is deeply embedded in the fabric of the city, with many of the buildings in the historic Old Town belonging to the university...
with an MA
Master of Arts (Scotland)
A Master of Arts in Scotland can refer to an undergraduate academic degree in humanities and social sciences awarded by the ancient universities of Scotland – the University of St Andrews, the University of Glasgow, the University of Aberdeen and the University of Edinburgh, while the University of...
in 1943.
His academic career was interrupted in World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
when he was involved in cryptography
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
at Bletchley Park
Bletchley Park
Bletchley Park is an estate located in the town of Bletchley, in Buckinghamshire, England, which currently houses the National Museum of Computing...
. After the war he became a research student of Louis Mordell
Louis Mordell
Louis Joel Mordell was a British mathematician, known for pioneering research in number theory. He was born in Philadelphia, USA, in a Jewish family of Lithuanian extraction...
at Trinity College
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...
, 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...
; he received his PhD
PHD
PHD may refer to:*Ph.D., a doctorate of philosophy*Ph.D. , a 1980s British group*PHD finger, a protein sequence*PHD Mountain Software, an outdoor clothing and equipment company*PhD Docbook renderer, an XML renderer...
in 1949 and was elected a fellow of Trinity in the same year.
After finishing his major, Cassels spent a year lecturing in mathematics 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...
before returning to Cambridge as a lecturer in 1950. He was appointed Reader
Reader (academic rank)
The title of Reader in the United Kingdom and some universities in the Commonwealth nations like Australia and New Zealand denotes an appointment for a senior academic with a distinguished international reputation in research or scholarship...
in Arithmetic in 1963, the same year he was elected as a fellow of the Royal Society of London. In 1967 he was appointed as Sadleirian Professor of Pure Mathematics at Cambridge. In 1969 he became Head of the Department of Pure Mathematics and Mathematical Statistics
Faculty of Mathematics, University of Cambridge
The Faculty of Mathematics at the University of Cambridge comprises the Department of Pure Mathematics and Mathematical Statistics and the Department of Applied Mathematics and Theoretical Physics . It is housed in the Centre for Mathematical Sciences site in West Cambridge, alongside the Isaac...
. He retired in 1984.
Mathematical work
He initially worked on elliptic curveElliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s. After a period when he worked on geometry of numbers
Geometry of numbers
In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
and diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
, he returned in the later 1950s to the arithmetic of elliptic curves, writing a series of papers connecting the Selmer group with Galois cohomology
Galois cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...
and laying some of the foundations of the modern theory of infinite descent
Infinite descent
In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the fact that the natural numbers are well ordered. One typical application is to show that a given equation has no solutions. Assuming a solution exists, one shows that another exists, that...
. His best-known single result may be the proof that the Tate-Shafarevich group, if it is finite, must have order that is a square; the proof being by construction of an alternating form.
Cassels has often studied individual Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
s by algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
and p-adic methods.
His publications include 200 papers. His advanced textbooks have influenced generations of mathematicians; some of Cassels's books have remained in print for decades.