Alan Baker (19 August 1939 – 4 February 2018) was an English

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

, known for his work on effective methods in number theory, in particular those arising from

transcendental number theory Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendenc ...

Life

Alan Baker was born in London on 19 August 1939. He attended

Stratford Grammar School Stratford may refer to: People * Stratford (surname), a list of people with the surname Stratford or de Stratford * Stratford (given name), a list of people * House of Stratford, a British aristocratic family * Tony Banks, Baron Stratford (1942– ...

, East London, and his academic career started as a student of

Harold Davenport Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory. Early life and education Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accringto ...

, at

University College London University College London (Trade name, branded as UCL) is a Public university, public research university in London, England. It is a Member institutions of the University of London, member institution of the Federal university, federal Uni ...

and later 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 ...

, where he received his PhD. 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 1970 when he was awarded the

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

at the age of 31. In 1974 he was appointed Professor of Pure Mathematics at

Cambridge University 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 ...

, a position he held until 2006 when he became an

Emeritus ''Emeritus/Emerita'' () is an honorary title granted to someone who retires from a position of distinction, most commonly an academic faculty position, but is allowed to continue using the previous title, as in "professor emeritus". In some c ...

. He was a fellow of Trinity College from 1964 until his death. His interests were in number theory, transcendence,

linear forms in logarithms In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. Nearly fifteen years earlier, Alexander Gelfond had considered the pr ...

, effective methods,

Diophantine geometry In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...

and

Diophantine analysis ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation *Diophantine equation *Diophantine quintuple In number theory, a diophantine -tuple is a set of p ...

. In 2012 he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

. He has also been made a foreign fellow of

the National Academy of Sciences, India The National Academy of Sciences, India, also known as NASI, was established in 1930 and is the oldest science academy of India. It is located in Prayagraj, Uttar Pradesh. Prof. Meghnad Saha was the founder president. Fellows *Suddhasatwa Basu ...

Research

Baker generalised the

Gelfond–Schneider theorem In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. History It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider. Statement Comments The values o ...

, which itself is a solution to

Hilbert's seventh problem Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers (''Irrationalität und Transzendenz bestimmter Zahlen''). Statement of the p ...

. Specifically, Baker showed that if

\alpha_1,...,\alpha_n

are

algebraic number In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...

s (besides 0 or 1), and if

\beta_1,..,\beta_n

are

irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...

algebraic numbers such that the set

\

linearly independent In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concep ...

over the rational numbers, then the number

\alpha_1^\alpha_2^\cdots\alpha_n^

is transcendental. Baker made significant contributions to several areas in number theory, such as the Gauss

class number problem In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each ''n'' ≥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers ''d'') having ...

, diophantine approximation, and to Diophantine equations such as the

Mordell curve In algebra, a Mordell curve is an elliptic curve of the form ''y''2 = ''x''3 + ''n'', where ''n'' is a fixed non-zero integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural ...

Selected publications

* * * *; *

Honours and awards

*1970:

*1972:

Adams Prize The Adams Prize is a prize awarded each year by the Faculty of Mathematics at St John's College to a UK-based mathematician for distinguished research in mathematical sciences. The prize is named after the mathematician John Couch Adams and wa ...

*1973: Fellowship 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 ...

References

External links

* * * {{DEFAULTSORT:Baker, Alan 1939 births 2018 deaths 20th-century English mathematicians 21st-century English mathematicians Fields Medalists British number theorists Alumni of University College London Alumni of Trinity College, Cambridge Fellows of Trinity College, Cambridge Fellows of the American Mathematical Society Fellows of the Royal Society Foreign fellows of the Indian National Science Academy Institute for Advanced Study visiting scholars Cambridge mathematicians Mathematicians from London