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Alan Baker

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19 Aug 1939

London, England

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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 at 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 A Thue who made the breakthrough to general results by proving in 1909 that all Diophantine equations of the form

f(x, y) = m

where m is an integer and f 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 f(x,y) = m described above there was a bound B which depended only on m and the integer coefficients of f with

max(|x0|, |y0|) B

for any solution (x0, y0) of f(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) and A concise introduction to the theory of numbers (1984). He also edited the important New advances in transcendence theory (1988).

Outside of mathematics, Baker lists his interests as travel, photography and the theatre.

Source:School of Mathematics and Statistics University of St Andrews, Scotland