恩里科·邦别里Enrico Bombieri

恩里科·邦别里年轻时便对数学产生了兴趣。在[2]中，作者们写道：-

与其他许多数学家一样，邦别里在相当早的年龄就对数学产生了兴趣。例如，13岁时，他正在学习数论中的一本教科书。

邦别里在米兰跟随吉奥凡尼·里奇学习，之后前往剑桥大学三一学院，在那里跟随哈罗德·达文波特学习。

邦别里因其在1974年于温哥华举行的国际数学家大会上的杰出工作而获得了菲尔兹奖。该奖项是为了表彰他对素数的研究、对单叶函数和局部路德维希·比贝尔巴赫猜想的研究、对多复变函数论以及对偏微分方程和极小曲面理论的主要贡献。特别是因为他在高维谢尔盖·伯恩斯坦问题上的工作。

Chandrasekharan在[3]中描述了邦别里在素数分布、单叶函数与局部路德维希·比贝尔巴赫猜想以及多复变函数方面的贡献。他写道：-

在邦别里的成就中，首要的是他在arithmetical progressions中关于素数分布的卓越定理，该定理是通过应用大筛法的方法得到的。

大筛法由尤里·弗拉基米罗维奇·林尼克于1941年引入，当时他试图解决伊万·维诺格拉多夫提出的问题。给定一个等差数列，大筛法给出关于任意有限整数集的分布的信息。雷尼·奥尔弗雷德在1950年进一步发展了尤里·弗拉基米罗维奇·林尼克的大筛法。然后，在1965年，克劳斯·罗特和邦别里独立地改进了雷尼·奥尔弗雷德的结果。邦别里应用他改进的大筛法证明了现在所谓的“邦别里均值定理”，该定理涉及等差数列中素数的分布。

1966年，邦别里被任命为比萨大学的数学讲席。他开始对De 恩尼奥·德乔吉及其几何测度论学派在比萨高等师范学校研究的问题产生兴趣。他们感兴趣的是三维以上空间的Plateau type problems。让我们指出他们正在研究的问题类型。

此处一段未译出，以下为英文原文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.

与约瑟夫·普拉托问题相对的是唯一性问题，上述卓越工作对此也有影响。1914年，谢尔盖·伯恩斯坦证明了三维欧几里得空间中形式为$f: \mathbb{R}^{2} \to \mathbb{R}$的极小曲面是一个平面。1965年，这一结果被de 德乔吉等人推广到具有$n ≤ 8$的$n$维欧几里得空间。他们证明了，对于$n ≤ 8$，形式为$f: \mathbb{R}^{n-1} \to \mathbb{R}$的极小超曲面是一个超平面。邦别里构造了例子来表明在$\mathbb{R}^{9}$中存在一个函数$f: \mathbb{R}^{8} \to \mathbb{R}$，它是$\mathbb{R}^{9}$中的极小曲面，但不是超平面。

[2]的作者们如下描述邦别里的能力：-

他一再展现出能够迅速掌握一个复杂新领域的要点，选择可接近的重要问题，并投入巨大的精力和洞察力去解决它们，同时大量利用其他数学家在迥然不同领域中的深刻结果。他的数学知识的广度对那些了解他及其工作的人来说是显而易见的。他也是一位优秀的数学写作者，他的讲座……因其清晰性而受到认可，这种清晰性随着所解释的数学思想的精妙程度而增加。

Chandrasekharan 在 [3] 中写道：-

…… 邦别里 的多才多艺与实力相结合，创造了许多既丰富又富有启发性的原创思想模式。

邦别里 于1980年被授予巴尔赞国际奖。邦别里 于1984年当选为法国 科学院 的外籍会员。文章 [4] 描述了 邦别里 的工作，正是这些工作促成了他的当选。

邦别里 目前在美国工作。1996年，邦别里 当选为 国家科学院 的会员。给他的引文如下：-

邦别里 是世界上最博学多才、最杰出的数学家之一。他对数论、代数几何、偏微分方程、多复变函数以及有限 群 理论产生了重大影响。他非凡的技术实力与对数学关键领域中核心问题的准确直觉相得益彰。

除上述奖项外，邦别里于1976年获得费尔特里内利奖，2002年获得意大利共和国功绩大军官勋章，2006年获得意大利克罗托内市颁发的国际毕达哥拉斯奖。2008年1月，在圣地亚哥举行的美国数学会第114届年会上，邦别里与Walter Gubler共同获得了2008年杜布奖。该奖项授予由邦别里和Gubler合著的Heights in Diophantine Geometry一书。奖项的评语如下：-

这本书是一部研究专著，涵盖丢番图几何的各个方面，既从算术几何的角度，也从超越数论的角度。……人们会感到，每一条引理、每一个定理、每一句评注都经过了仔细斟酌，每一个证明的每一个细节都经过了深思熟虑。每一章中都有精心挑选、富有启发性的例子。这本书在其独创的方法、无与伦比的全面性以及纯粹的阐述优雅性方面都是一部杰作。毫无疑问，这本书将成为现代数学这一核心学科未来发展的基础。

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.