MathematicianItalianFields MedalistAcademician

Enrico Bombieri

Influential Italian mathematician Enrico Bombieri is recognized for contributions to analytic number theory, Diophantine geometry, and the large sieve, earning the Fields Medal in 1974.

1. Overview

Enrico Bombieri is an influential Italian mathematician born on November 26, 1940, in Milan, Italy. He is widely recognized for his profound contributions across various fields of mathematics, including analytic number theory, Diophantine geometry, complex analysis, group theory, minimal surfaces, and partial differential equations. His groundbreaking work, particularly on the large sieve method and its application to the distribution of prime numbers, earned him the prestigious Fields Medal in 1974, the highest honor in mathematics. Bombieri currently holds the position of professor emeritus in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey, where he is also known as the IBM von Neumann Professor. His intellectual contributions have significantly advanced mathematical understanding and continue to impact the field.

2. Early Life and Education

Enrico Bombieri's formative years laid the groundwork for his distinguished career in mathematics, beginning with his early intellectual curiosity and culminating in advanced studies at renowned institutions.

2.1. Birth and Early Life

Enrico Bombieri was born in Milan, Italy, on November 26, 1940. His mathematical aptitude became evident at an early age, as he published his first mathematical paper in 1957 when he was just 16 years old.

2.2. Education

Bombieri pursued his higher education in mathematics, earning his Laurea (first degree) from the University of Milan in 1963 at the age of 22. His doctoral studies were supervised by Giovanni Ricci. Following his studies in Milan, he continued his academic journey at Trinity College, Cambridge, where he undertook postgraduate work under the guidance of the eminent mathematician Harold Davenport. This period of intensive training and collaboration with leading figures in the field prepared him for his future contributions to mathematics.

3. Career

Enrico Bombieri's professional trajectory as a mathematician spans significant tenures at prestigious academic institutions in both Italy and the United States.

3.1. Career in Italy

After completing his studies, Bombieri began his academic career in Italy. He served as an assistant professor at the University of Cagliari from 1963 to 1965, subsequently being promoted to a full professorship there from 1965 to 1966. In 1966, he moved to the University of Pisa, where he held a professorship until 1974. Following this, he was appointed professor at the Scuola Normale Superiore di PisaSuperior Normal School of PisaItalian, a position he held from 1974 to 1977.

3.2. Career in the United States

In 1977, Bombieri transitioned his career to the United States. He became a professor in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey. He remained a distinguished faculty member at the Institute for many years, eventually becoming professor emeritus in 2011. During his tenure at the Institute for Advanced Study, he also held the esteemed title of IBM von Neumann Professor.

4. Major Research and Achievements

Enrico Bombieri's research has profoundly impacted several branches of mathematics, characterized by his innovative methods and the resolution of long-standing problems.

4.1. Analytic Number Theory

Bombieri is particularly renowned for his foundational work in analytic number theory. His development of the large sieve method is considered a cornerstone of modern number theory. A major application of this method is the Bombieri-Vinogradov theorem, which significantly refines Dirichlet's theorem on arithmetic progressions. This theorem demonstrates that, by averaging over the modulus within a specific range, the mean error in the distribution of prime numbers in arithmetic progressions is substantially smaller than what can be proven for individual cases. The power of this result is such that it can, in certain contexts, serve as a viable substitute for the still-unproven generalized Riemann hypothesis. The concept of the large sieve was originally introduced by Yuri Linnik in 1941.

4.2. Diophantine Geometry

In Diophantine geometry, Bombieri's research has focused on problems concerning integer solutions to polynomial equations and their geometric interpretations. His work has provided crucial insights into the intricate relationships between number theory and algebraic geometry.

4.3. Complex Analysis and Group Theory

Bombieri has also made significant contributions to the fields of complex analysis and group theory. His work includes the development or elucidation of key theorems and concepts. Notably, his research extends to multi-variable complex function theory and includes his resolution of the local Bieberbach conjecture for univalent functions, which addressed a significant problem in complex analysis.

4.4. Minimal Surfaces and Partial Differential Equations

In the area of minimal surfaces and related partial differential equations, Bombieri achieved a notable breakthrough in 1969. Collaborating with Ennio De Giorgi and Enrico Giusti, he successfully resolved Bernstein's problem in dimensions above eight. This achievement addressed a fundamental question regarding the properties of minimal surfaces in higher-dimensional spaces.

4.5. Asymptotic Sieve

In 1976, Bombieri introduced and further developed a mathematical technique known as the "asymptotic sieve." This method has proven to be a valuable tool in various number theoretic investigations.

4.6. Finite Simple Groups

Bombieri's contributions extend to the monumental project of the classification of finite simple groups. In 1980, he provided the final completion of the proof regarding the uniqueness of finite groups of Ree type in characteristic 3. At the time of its publication, this was a critical missing step in the comprehensive classification of all finite simple groups.

5. Awards and Honors

Enrico Bombieri's exceptional contributions to mathematics have been recognized with numerous prestigious awards, prizes, and academic honors from around the world.

5.1. Fields Medal

The most significant recognition of Bombieri's work came in 1974 when he was awarded the Fields Medal. This honor, widely regarded as the highest accolade in the field of mathematics, was bestowed upon him for his groundbreaking research on the large sieve and its profound applications to understanding the distribution of prime numbers. In the same year, he was also invited as a plenary speaker at the International Congress of Mathematicians held in Vancouver.

5.2. Other Major Awards

Beyond the Fields Medal, Bombieri has received a series of other distinguished international awards. These include the Caccioppoli Prize in 1966, the Balzan Prize in 1980, and the Pythagoras Prize in 2006. In 2010, he was a joint recipient of the King Faisal International Prize in the Science category, sharing the honor with Terence Tao. Most recently, in 2020, he was awarded the Crafoord Prize in Mathematics, further underscoring the breadth and lasting impact of his work.

5.3. Academy Memberships and Honors

Bombieri's scientific eminence is also reflected in his election to several prestigious academic societies and academies globally. He was elected to the Accademia Nazionale dei LinceiNational Academy of the LinceiItalian in 1976, followed by the French Academy of Sciences in 1984, the Academia Europaea in 1995, and the United States National Academy of Sciences in 1996. In recognition of his contributions to Italy, he was made a Cavaliere di Gran Croce al Merito della Repubblica ItalianaKnight Grand Cross of the Order of Merit of the Italian RepublicItalian in 2002.

6. Contributions to the Mathematics Community

Enrico Bombieri has demonstrated a strong commitment to the broader mathematics community through his dedicated pro bono service. He has actively served on external review boards, contributing to the rigorous evaluation of academic programs and research. Furthermore, he is known for his meticulous and thorough peer-reviewing of exceptionally complex manuscripts. A notable instance of this dedication was his review of the extensive paper by Per Enflo on the challenging invariant subspace problem.

7. Personal Interests

Beyond his profound contributions to mathematics, Enrico Bombieri maintains a diverse range of personal interests and hobbies. He is an accomplished artist and a serious painter, often carrying his paints and brushes with him during his travels. His artistic pursuits sometimes intersect with his mathematical world; for instance, one of his paintings depicts a giant chessboard by a lake, with the pieces arranged to recreate a critical moment from the historic match where IBM's chess-playing computer, Deep Blue, defeated Garry Kasparov. Bombieri himself was a member of the Cambridge University chess team. He is also known to be a gourmet cook. In his younger years, he indulged his interest in botany by exploring the Alps, where he would search for wild orchids and other plants.

8. Selected Publications

This section provides a curated list of Enrico Bombieri's most influential scholarly works, encompassing both solo and collaborative publications that have significantly shaped mathematical discourse.

  • Sole Publications**:
    • Le Grand Crible dans la Théorie Analytique des Nombres (Seconde Édition). Astérisque 18, Paris 1987.
    • Joint Publications**:
      • "On Siegel's lemma" with J. Vaaler, Inventiones Mathematicae, Volume 73, Issue 1, pages 11-32, February 1983.
      • "On effective measures of irrationality for roots and related numbers" with J. Mueller, Journal für die reine und angewandte Mathematik, Volume 342, pages 173-196, 1983.
      • "Product of polynomials in many variables" with B. Beauzamy, Per Enflo, and H. L. Montgomery, Journal of Number Theory, pages 219-245, 1990.
      • Heights in Diophantine Geometry with Walter Gubler, Cambridge U. P., 2006.