# Enrico Bombieri

### Quick Info

Born
26 November 1940
Milan, Italy

Summary
Enrico Bombieri is an Italian Fields medal winner, known for his work in analytic number theory, algebraic geometry, theory of several complex variables, minimal surfaces and the theory of finite groups.

### Biography

Enrico Bombieri became interested in mathematics when he was young. In [2] the authors write:-
Like a number of other mathematicians, Bombieri became interested in mathematics at a fairly early age. At 13, for example, he was studying a textbook in number theory.
Bombieri studied with G Ricci in Milan and then went to Trinity College, Cambridge where he studied with H Davenport.

Bombieri was awarded a Fields Medal for his outstanding work at the International Congress of Mathematicians held in Vancouver in 1974. The award was made for his major contributions to the study of the prime numbers, to the study of univalent functions and the local Bieberbach conjecture, to the theory of functions of several complex variables, and to the theory of partial differential equations and minimal surfaces. In particular for his work on Sergei Bernstein's problem in higher dimensions.

Chandrasekharan in [3] describes Bombieri's contributions to the distribution of primes, to univalent functions and the local Bieberbach conjecture and to functions of several complex variables. He writes:-
First among Bombieri's achievements is his remarkable theorem on the distribution of primes in arithmetical progressions, which is obtained by an application of the methods of the large sieve.
The large sieve method was introduced by Linnik in 1941 in his attempts to solve problems posed by Vinogradov. Given an arithmetic progression, the large sieve gives information about the distribution of an arbitrary finite set of integers. Rényi developed Linnik's large sieve methods further in 1950. Then, in 1965, Klaus Roth and Bombieri independently sharpened Rényi's results. Bombieri applied his improved large sieve method to prove what is now called "Bombieri's mean value theorem", which concerns the distribution of primes in arithmetic progressions.

In 1966 Bombieri was appointed to a chair of mathematics at the University of Pisa. He began to become interested in problems that De Giorgi and his school of geometric measure theory were working on at the Scuola Normale Superiore in Pisa. They were interested in Plateau type problems for spaces of more than three dimensions. Let us indicate the type of problems they were studying.

For high-dimensional Euclidean space they were investigating the minimal varieties of the family of submanifolds. These minimal varieties generalise the minimal surfaces in the Plateau problem. The meaning of minimal for a $k$-dimensional submanifold $M$ of an $n$-dimensional space is that a sufficiently small piece of $M$ has the least volume compared with other $k$-dimensional submanifolds $M'$ where $M$ and $M'$ have the same $(k - 1)$-dimensional boundary. A minimal hypersurface, that is a submanifold with $k = n - 1$, with a given boundary had been shown not to contain singular points for $n ≤ 7$. Bombieri, working with de Giorgi and Giusti, proved in 1969 that for $n ≥ 8$ there is a minimal hypersurface with an essential singularity.

In contrast to the Plateau problem is the uniqueness problem and the remarkable work described above had implications for this too. In 1914 Sergei Bernstein had proved that a minimal surface in 3-dimensional Euclidean space of the form $f: \mathbb{R}^{2} \to \mathbb{R}$, is a plane. In 1965 this result had been extended by de Giorgi and others to $n$-dimensional Euclidean spaces with $n ≤ 8$. They proved that, for $n ≤ 8$, a minimal hypersurface of the form $f: \mathbb{R}^{n-1} \to \mathbb{R}$ is a hyperplane. Bombieri constructed examples to show that in $\mathbb{R}^{9}$ there is a function $f: \mathbb{R}^{8} \to \mathbb{R}$ which is a minimal surface in $\mathbb{R}^{9}$ which is not a hyperplane.

The authors of [2] describe Bombieri abilities as follows:-
He has repeatedly demonstrated an ability to quickly master essentials of a complicated new field, to select important problems which are accessible, and to apply intense energy and insight to their solution, making liberal use of deep results of other mathematicians in widely differing areas. The breadth of his mathematical knowledge is clearly visible to those who know him and his work. He is also a fine writer of mathematics, and his lectures ... are recognised for clarity which increases with the subtlety of the mathematical idea being explained.
Chandrasekharan, in [3], writes:-
... Bombieri's versatility and strength have combined to create many original patterns of ideas which are both rich and inspiring.
Bombieri was awarded the Balzan International Prize in 1980. Bombieri was elected a foreign member of the French Academy of Sciences in 1984. The article [4] describes Bombieri's work which led to his election.

Bombieri now works in the United States. In 1996 Bombieri was elected to membership of the National Academy of Sciences. The citation for him read:-
Bombieri is one of the world's most versatile and distinguished mathematicians. He has significantly influenced number theory, algebraic geometry, partial differential equations, several complex variables, and the theory of finite groups. His remarkable technical strength is complemented by an unerring instinct for the crucial problems in key areas of mathematics.
In addition to the awards mentioned above, Bombieri received the Feltrinelli Prize in 1976, the Cavaliere di Gran Croce al Merito della Repubblica, Italy in 2002, and the Premio Internazionale Pitagora from the City of Crotone in Italy in 2006. Jointly with Walter Gubler, Bombieri was awarded the 2008 Doob Prize at the 114th Annual Meeting of the American Mathematical Society in San Diego in January 2008. The award was made for the book Heights in Diophantine Geometry jointly authored by Bombieri and Gubler. The citation for the prize reads:-
The book is a research monograph on all aspects of Diophantine geometry, both from the perspective of arithmetic geometry and of transcendental number theory. ... One gets the sense that every lemma, every theorem, every remark has been carefully considered, and every proof has been thought through in every detail. There are well-chosen illuminating examples throughout every chapter. The book is a masterpiece in terms of its original approach, its unrivalled comprehensiveness, and the sheer elegance of the exposition. There can be no doubt that this book will become the basis for the future development of this central subject of modern mathematics.

### References (show)

1. Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Enrico-Bombieri
2. F J Almgren and H Montgomery, The 1974 Fields medals II : An analyst and number theorist, Science 186 (4159) (1974), 130-131.
3. K Chandrasekharan, The work of Enrico Bombieri, Proceedings of the International Congress of Mathematicians, Vancouver 1974 1 (Montreal, Que., 1975), 3-10.
4. Enrico Bombieri (French), C. R. Acad. Sci. Sér. Gén. Vie Sci. 1 (6) (1984), 513.
5. W Schwarz, Zum zahlentheoretischen Werk von Enrico Bombieri, Jahrbuch Überblicke Mathematik, 1975 (Mannheim, 1975), 150-151.

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Honours awarded to Enrico Bombieri

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Written by J J O'Connor and E F Robertson
Last Update September 2009