# Alan Baker

### Quick Info

Born
19 August 1939
London, England
Died
4 February 2018
Cambridge, England

Summary
Alan Baker was an English mathematician, known for his work in number theory.

### Biography

Alan Baker was educated at Stratford Grammar School. From there, after winning a State Scholarship, he entered University College London where he studied for his B.Sc. He was awarded a B.Sc. with First Class Honours in Mathematics in 1961. He moved on to Trinity College Cambridge where he was awarded an M.A. Continuing his research at Cambridge advised by Harold Davenport, Baker began publishing papers. In fact eight of his papers had appeared in print before he submitted his doctoral dissertation: Continued fractions of transcendental numbers (1962); On Mahler's classification of transcendental numbers (1964); Rational approximations to certain algebraic numbers (1964); On an analogue of Littlewood's Diophantine approximation problem (1964); Approximations to the logarithms of certain rational numbers (1964); Rational approximations to the cube root of 2 and other algebraic numbers (1964); Power series representing algebraic functions (1965); and On some Diophantine inequalities involving the exponential function (1965). He received his doctorate from the University of Cambridge for his thesis Some Aspects of Diophantine Approximation in 1965. In the same year he was elected a Fellow of Trinity College. He spent the academic year 1964-65 at the Department of Mathematics, University College London.

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 his career at Cambridge 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. He also a visiting professor at the University of Hong Kong in 1988, at the Eidgenössische Technische Hochschule Zürich in 1989, and at the Mathematical Sciences Research Institute, Berkeley, California in 1993.

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 Paul Turán in [11], who first gives the historical setting:-
The theory of transcendental numbers, initiated by Liouville in 1844, has been enriched greatly in recent years. Among the relevant profound contributions are those of Alan Baker, Wolfgang M Schmidt, and Vladimir Gennadievich Sprindzuk. Their work moves in important directions which contrast with the traditional concentration on the deep problem of finding significant classes of finding functions assuming transcendental values for all non-zero algebraic values of the independent variable. Among these, Baker's have had the heaviest impact on other problems in mathematics. Perhaps the most significant of these impacts has been the application to Diophantine equations. This theory, 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 $f (x, y) = m$ 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
$max(|x_{0}|, |y_{0}|) ≤ B$
for any solution $(x_{0}, y_{0})$ 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 $a^{q}$ 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 Aleksandr Gelfond and Theodor Schneider in 1934 but Baker ([6]):-
... 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 [11] 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.
Among Baker's famous books are: Transcendental number theory (1975), Transcendence theory : advances and applications (1977), A concise introduction to the theory of numbers (1984), (with Gisbert Wüstholz) Logarithmic forms and Diophantine geometry (2007), and A Comprehensive Course in Number Theory (2012). We quote from the introduction to Transcendental number theory (1975):-
The study of transcendental numbers... has now developed into a fertile and extensive theory, enriching widespread branches of mathematics. My aim has been to provide a comprehensive account of the recent major discoveries in the field. Classical aspects of the subject are discussed in the course of the narrative. Proofs in the subject tend... to be long and intricate, and thus it has been necessary to select for detailed treatment only the most fundamental results; moreover, generally speaking, emphasis has been placed on arguments which have led to the strongest propositions known to date or have yielded the widest application.
Robert Tijdeman writes in a review of this book:-
The author has succeeded in his plan. This book gives a survey of the highlights of transcendental number theory, in particular of the author's own important contributions for which he was awarded a Fields medal in 1970. It is a very useful publication for mathematicians who want to obtain a general insight into transcendence theory, its techniques and its applicability. The style is extremely condensed, but there are many references for more detailed study. The presentation is very well done.
This book is also reviewed by Heini Halberstam (1926-2014) who writes [7]:-
Within the space of a mere 130 pages the author gives a panoramic account of modern transcendence theory, based on his own Adams Prize essay. The fact that this is now "a fertile and extensive theory, enriching wide-spread branches of mathematics" is due in large measure to the author himself, who was awarded in 1970 a Fields Medal (the Nobel Prize of mathematics) for his contributions. The prose is clear and economical yet interspersed with flashes of colour that convey a sense of personality; and each chapter begins with a helpful summary of the subsequent matter. The mathematical argument at all stages is highly condensed, as, indeed, is inevitable in a short research monograph covering so much ground. One might reproach the author for not having been more merciful to the beginner; but even a beginner can gain from the book a clear impression of what are the major achievements to date in this profoundly difficult field and which are the outstanding problems, while for others there is here a wealth of material for numerous fruitful study-groups.
Don Redmond, reviewing Baker's 1984 "Concise Introduction", writes:-
Many books do not live up to their titles, but this is one that definitely does. The book is very concise and would be a nice reference, since it covers the key points of a standard course, but the reviewer is not sure that one could use it as the sole textbook of a first course in number theory.
David Singmaster, also reviewing the "Concise Introduction", writes [10]:-
Introductions to number theory are numerous, so any new introduction must be examined for novelty. This book is the material for a lecture course at the University of Cambridge. Consequently, "concise" is no exaggeration. ... Overall, the book is a marvel of condensation. This would be true even if all 91 pages of text were devoted to the main material, but he has condensed further and uses about 30 pages for his supplementary material. This contains the most useful summary of current number theory that I have seen. There is a competent index so one can locate the results. ... I would recommend this book to any serious undergraduate wanting a survey of the field, but I would warn him that the proofs require close attention. Anyone with some background in number theory will highly appreciate Baker's exposition of current knowledge
Yuri Bilu states in a review of Baker and Wüstholz's 2007 book Logarithmic forms and Diophantine geometry:-
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. ... This book is indeed an introduction. Its purpose is to teach principles while avoiding technicalities. This imposes certain limitations on the content. The authors treat in great detail the qualitative theory for the multiplicative group, but do not say much on the quantitative aspect, and only briefly mention abelian varieties. However, this book gives the necessary intuitive background to study the original journal articles of Baker, Masser, Wüstholz and others on the above-listed subjects.
Baker also edited the important New advances in transcendence theory (1988) and wrote the important survey with Gisbert Wüstholz entitled Number theory, transcendence and Diophantine geometry in the next millennium. This is a survey of achievements and open problems in transcendence theory and related mathematics.

In 1999 a conference was organised in Zürich 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 Zürich 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.
In [5] Baker makes remarks on the history of number theory, in particular on transcendental numbers. We quote from his paper:-
Well, what does this tell us about the historical evolution of mathematics? First it is clear that a very important role has been played by a few key problems, centres of attraction, in Professor Dieudonné's terminology. This may be more true of number theory than other branches of mathematics but I believe that all good work has been guided to some extent by such centres. The general trend of the particular field that I have been discussing is difficult to summarise, since it has involved in its development many novel twists and turns; but one obvious element in the evolution has been the successful blending, or fusion, of ideas from number theory and algebra with the progressively wider use of classical function theory. And it is this convergence of diverse concepts that forms the essential ingredient, I believe, in the creation of an active theory. According to Professor Dieudonné, the study of transcendental numbers is only just on its way to becoming a "method". Given, however, the diverse nature of the problems which it has been instrumental in solving, there seems little doubt that it reached the latter stage several years ago, and it would appear, in fact that it is already on the path of becoming, in Professor Dieudonné's language, a centre of radiation.
Here are Baker's research interests as given on his University of Cambridge page [9] (consulted in January 2014):-
Baker's Theorem on the linear independence of logarithms of algebraic numbers has been the key to a vast range of developments in number theory over the past thirty years. Amongst the most significant are applications to the effective solution of Diophantine equations, to the resolution of class-number problems, to the theory of p-adic L-functions and especially, through works of Masser and Wüstholz, to many deep aspects of arithmetical algebraic geometry. The theory continues to be a source of much fruitful research to the present day.
Baker has received many honours for his mathematical contributions in addition to the 1970 Fields medal. These include the award of the Adams prize from the University of Cambridge (1972) and election to the Royal Society of London (1973). He was awarded an honorary doctorate from Université Louis Pasteur Strasbourg (1998), made an honorary fellow of University College London (1979), a foreign fellow of the Indian Academy of Science (1980), foreign fellow of the National Academy of Sciences India (1993), a member of the Academia Europaea (1998), and an honorary member of the Hungarian Academy of Sciences (2001).

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

### References (show)

1. Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Alan-Baker
2. Alan Baker, Heidelberg Laureate Forum. http://www.heidelberg-laureate-forum.org/blog/laureate/alan-baker/
3. A Baker, Effective methods in the theory of numbers, Actes du Congrès International des Mathématiciens, Nice, 1970 Vol. 1 (Paris, 1971).
4. A Baker, Effective methods in the theory of numbers/Diophantine problems, in M Atiyah and D Iagolnitzer (eds.), Fields Medallists Lectures (World Sci. Publ., Singapore, 1997), 190-193.
5. A Baker, Some historical remarks on number theory, Historia Mathematica 2 (1975), 549-553.
6. Alan Baker, in M Atiyah and D Iagolnitzer (eds.), Fields Medallists Lectures (World Sci. Publ., Singapore, 1997), 161.
7. H Halberstam, Review: Transcendental Number Theory, by Alan Baker, The Mathematical Gazette 59 (410) (1975), 280-282.
8. J C Peral, Alan Baker: transcendental work (Spanish), Gac. R. Soc. Mat. Esp. 4 (2) (2001), 437-445.
9. Professor Alan Baker, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge (2014). https://www.dpmms.cam.ac.uk/people/ab10005/
10. D Singmaster, Review: A Concise Introduction to the Theory of Numbers, by Alan Baker, The Mathematical Gazette 69 (450) (1985), 318-319.
11. P Turán, On the work of Alan Baker, Actes du Congrès International des Mathématiciens, Nice, 1970 Vol. 1 (Paris, 1971), 3-5.
12. P Turán, On the work of Alan Baker, in M Atiyah and D Iagolnitzer (eds.), Fields Medallists Lectures (World Sci. Publ., Singapore, 1997), 157-159.