## The Laureates

### Alan Baker

Born 19 August 1939 in London (UK)

Fields Medal (1970) for his work in number theory especially in the areas of transcendence and Diophantine geometry.

He was educated at Stratford Grammar School and after receiving the award of a State Scholarship he began studies in mathematics at University College London graduating BSc with first class honours in 1961. He went on to Cambridge to take up research under the guidance of Harold Davenport, was elected a Fellow of Trinity College Cambridge in 1964 and obtained the PhD in 1965 with a dissertation in the area of Diophantine approximation. He spent the year 1964-65 as a Faculty member at the Department of Mathematics, University College London and subsequently returned to Cambridge becoming Professor of Pure Mathematics in the University with a personal chair from 1974 to 2006; he remains currently in Cambridge as Professor Emeritus and a Fellow of Trinity College. He has held visiting posts at various places including the Institute for Advanced Study Princeton (1970), Stanford University (1974), the University of Hong Kong (1988), ETH Zurich (1989) and MSRI Berkeley (1993). He was awarded the Adams Prize of the University of Cambridge in 1972, was elected a Fellow of the Royal Society of London in 1973, an honorary Fellow of University College London in 1979, a Foreign Fellow of the Indian National Science Academy (Delhi) in 1980 and of the National Academy of Sciences India (Allahabad) in 1993.  He received an honorary doctorate from the Université Louis Pasteur (Strasbourg) in 1998 and he became an honorary member of the Hungarian Academy of Sciences in 2001. He gave the first Paul Turán Memorial Lecture of the J. Bolyai Mathematical Society (Hungary) in 1978 and has lectured widely elsewhere; he has taken some notable research students who have themselves gone on to distinguished careers.

He is well known for the publication of several books including the monograph Transcendental number theory (1975). The subject has evolved from the classic 19th century studies of Hermite and Lindemann that culminated in a proof that $\pi$ is transcendental and gave at last a resolution to the notorious ancient Greek ‘squaring of the circle’ problem; another famous landmark dates from the 1930s when Alexander Gelfond (1906-1968) and Theodor Schneider (1911-1998) succeeded in showing that numbers of the form $\alpha^\beta$ with algebraic $\alpha$ and algebraic irrational $\beta$, such as 2 to the power $\sqrt{2}$, are transcendental and thereby answered the seventh of the celebrated list of problems that Hilbert had raised at the International Congress of 1900. Baker’s work of the 1960s greatly advanced the subject; he obtained a theorem which now bears his name about logarithms of algebraic numbers that went well beyond the results just mentioned and has solution of Diophantine equations, to the resolution of class number problems, to the theory of been the key to a vast range of developments in number theory with applications especially to the p-adic L-functions and to numerous deep aspects of arithmetical algebraic geometry.

Diophantine equations have fascinated people since ancient times. An equation is called ‘Diophantine’ if it is only its integer solutions that are of interest . Baker’s work threw important light on equations of the form $f(x,y)=m$ where m is an integer and f is a polynomial in two variables fulfilling certain conditions — mathematically, it is called a ‘homogeneous irreducible binary form of degree at least three’. As early as the beginning of the 20th century it became clear that such equations have only a finite number of solutions in integers; in other words, considering the equations as curves in the plane, they contain only a finite number of points $(s,t)$ with integers s and  t. But Baker in his study of these curves found more: he discovered that the values of s and  t cannot increase arbitrarily but must remain below a limit depending only on m and the coefficients of the form. Thus, in principle, the equation can be completely solved by a finite amount of computation — methods of investigation that lead to such an outcome have been dubbed ‘effective’ and Baker has become famous for his creation of effective methods of this kind; several mathematical software packages accessible through the internet crucially incorporate them.