Alan Baker, FRS (born on 19 August 1939) is an English mathematician. He was born in London. He is known for his work on effective methods in number theory, in particular those arising from transcendence theory. He was awarded the Fields Medal in 1970, at age 31. His academic career started as a student of Harold Davenport, at University College London and later at Cambridge. He is a fellow of Trinity College, Cambridge.
His interests are:
1. Number theory
3. Logarithmic forms
4. Effective methods
5. Diophantine geometry
6. Diophantine analysis
His students include John Coates, David Masser, Roger Heath-Brown, Yuval Flicker, and Cameron Stewart.
* Fellow Royal Society, Fellow Indian National Science Academy
Alan Baker was educated at Stratford Grammar School. From there he entered University College London where he studied for his B.Sc., moving on to Trinity College Cambridge where he was awarded an M.A. Continuing his research at Cambridge, Baker received his doctorate and was elected a Fellow of Trinity College in 1964.
From 1964 to 1968 Baker was a research fellow at Cambridge, then becoming Director of Studies in Mathematics, a post which he held from 1968 until 1974 when he was appointed Professor of Pure Mathematics. During this time he spent time in the United States, becoming a member of the Institute for Advanced Study in Princeton in 1970 and visiting professor at Stanford in 1974.
Baker was awarded a Fields Medal in 1970 at the International Congress at Nice. This was awarded for his work on Diophantine equations. This is described by Turán in , who first gives the historical setting:-
[Diophantine equations], carrying a history of more than one thousand years, was, until the early years of this century, little more than a collection of isolated problems subjected to ingenious ad hoc methods. It was Axel Thue who made the breakthrough to general results by proving in 1909 that all Diophantine equations of the form
ƒ(x, y) = m
where m is an integer and ƒ is an irreducible homogeneous binary form of degree at least three, with integer coefficients, have at most finitely many solutions in integers.
Turán goes on to say that Carl Siegel and Klaus Roth generalised the classes of Diophantine equations for which these conclusions would hold and even bounded the number of solutions. Baker however went further and produced results which, at least in principle, could lead to a complete solution of this type of problem. He proved that for equations of the type ƒ(x, y) = m described above there was a bound B which depended only on m and the integer coefficients of ƒ with
max(|x0|, |y0|) ≤ B
for any solution (x0, y0) of ƒ(x, y) = m. Of course this means that only a finite number of possibilities need to be considered so, at least in principle, one could determine the complete list of solutions by checking each of the finite number of possible solutions.
Baker also made substantial contributions to Hilbert's seventh problem which asked whether or not aq was transcendental when a and q are algebraic. Hilbert himself remarked that he expected this problem to be harder than the solution of the Riemann conjecture. However it was solved independently by Gelfond and Schneider in 1934 but Baker ():-
... succeeded in obtaining a vast generalisation of the Gelfond-Schneider Theorem ... From this work he generated a large category of transcendental numbers not previously identified and showed how the underlying theory could be used to solve a wide range of Diophantine problems.
Turán  concludes with these remarks:-
I remark that [Baker's] work exemplifies two things very convincingly. Firstly, that beside the worthy tendency to start a theory in order to solve a problem it pays also to attack specific difficult problems directly. ... Secondly, it shows that a direct solution of a deep problem develops itself quite naturally into a healthy theory and gets into early and fruitful contact with significant problems of mathematics.
In addition to the honour of the Fields Medal, Baker has received many other honours including the Adams Prize from the University of Cambridge in 1972 and election to become a Fellow of the Royal Society in 1973. In 1980 he was elected an honorary Fellow of the Indian National Science Academy.
Among his famous books are Transcendental number theory (1975), Transcendence theory : advances and applications (1977), A concise introduction to the theory of numbers (1984) and (with Gisbert Wüstholz) Logarithmic forms and Diophantine geometry (2007). Yuri Bilu begins a review of this last mentioned work as follows:-
This long-awaited book is an introduction to the classical work of Baker, Masser and Wüstholz in a form suitable for both undergraduate and graduate students.
Baker also edited the important New advances in transcendence theory (1988).
In 1999 a conference was organised in Zurich to celebrate Baker's 60th birthday. Most of the lectures given at the meetings were published in A Panorama in Number Theory or The View from Baker's Garden (2002). The introduction to the book begins as follows:-
The millennium, together with Alan Baker's 60th birthday offered a singular occasion to organize a meeting in number theory and to bring together a leading group of international researchers in the field; it was generously supported by ETH Zurich together with the Forschungsinstitut für Mathematik. This encouraged us to work out a programme that aimed to cover a large spectrum of number theory and related geometry with particular emphasis on Diophantine aspects. ... The London Mathematical Society was represented by its President, Professor Martin Taylor, and it sent greetings to Alan Baker on the occasion of his 60th birthday.
Outside of mathematics, Baker lists his interests as travel, photography and the theatre.
* O'Connor, John J.; Robertson, Edmund F., "Alan Baker (mathematician)", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Baker_Alan.html .
* Alan Baker (mathematician) at the Mathematics Genealogy Project
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