Enrico Bombieri

fl. 1940, Milan, Italy

Arithmetic and number theoryAnalysisGeometryFields Medal 1974

Italian-American mathematician working in number theory and analysis, known for the Bombieri–Vinogradov theorem on primes in progressions and for work on minimal surfaces; Fields Medal 1974.

Enrico Bombieri was born in Milan and published his first mathematical papers while still a teenager. He studied at the University of Milan and then at Cambridge, becoming professor at the University of Cagliari and later at Pisa and the Scuola Normale Superiore. In 1965 he proved the large-sieve mean-value theorem now known as the Bombieri–Vinogradov theorem, showing that the error in the distribution of primes over arithmetic progressions is small on average for moduli up to nearly the square root of the range — information comparable in strength, on average, to the generalized Riemann hypothesis.

His work of the same period spans Diophantine approximation and transcendence, the theory of univalent functions, and geometric measure theory: with De Giorgi and Giusti he proved that the Bernstein problem — whether entire minimal graphs in $\mathbb{R}^{n+1}$ are hyperplanes — has a negative answer for $n \geq 8$ , and he constructed the singular minimal cone that makes the bound sharp. He also made influential contributions to heights and Diophantine geometry, including work related to the Mordell conjecture and, later, to the effective results of the theory of $S$ -unit equations.

Bombieri moved to the Institute for Advanced Study in Princeton in 1977, where he spent the rest of his career. He received the Fields Medal in 1974. His later interests include the Riemann hypothesis, Selberg's sieve and problems at the interface of number theory and algebraic geometry.

Workplaces: Institute for Advanced Study, Princeton, University of Pisa, Scuola Normale Superiore, Pisa

Italy, United States

Award-winning work

Fields Medal 1974Awarded for contributions to number theory, analysis and the geometry of minimal surfaces. Bombieri developed the large sieve into a sharp mean-value theorem — the Bombieri–Vinogradov theorem (1965) — showing that primes are well distributed in arithmetic progressions on average beyond the reach of the generalized Riemann hypothesis, a result that powered advances such as bounded gaps between primes. He also proved, with Ennio De Giorgi and Enrico Giusti, that Bernstein's problem on minimal graphs fails in dimensions eight and above, linking minimal surfaces to sets of finite perimeter and singular solutions.

Contributions, linked to the library

TheoremBounded gaps between primes (Zhang, Maynard)The Bombieri–Vinogradov theorem is the level-of-distribution input behind Zhang's bounded gaps and Maynard's method.Great problemRiemann hypothesisThe large-sieve mean-value theorem supplies, on average, information of the strength the generalized Riemann hypothesis would give.Library topicSieve methodsDeveloped the large sieve into the Bombieri–Vinogradov mean-value theorem, a cornerstone of sieve theory.