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Enrico Bombieri

## Summary

Enrico Bombieri (born 26 November 1940, Milan) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory.[5] Bombieri is currently Professor Emeritus in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey.[6] Bombieri won the Fields Medal in 1974[5] for his contributions to large sieve mathematics,[7] conceptualized by Linnick 1941,[8][9] and its application to the distribution of prime numbers.[7]

Enrico Bombieri
Enrico Bombieri
Born26 November 1940 (age 81)
NationalityItalian
Alma materUniversity of Milan
Known forDeterminant method
Large sieve method in analytic number theory
Bombieri-Lang conjecture
Bombieri norm
"Heights" in Diophantine geometry
Siegel's lemma for bases (Bombieri–Vaaler)
Partial differential equations
Awards1966, Caccioppoli Prize[1]
1974, Fields Medal
1976, Feltrinelli Prize
1980, Balzan Prize
2006, Pythagoras Prize[2]
2008, Joseph L. Doob Prize[3][4]
2010, King Faisal International Prize
2020, Crafoord Prize
Scientific career
FieldsMathematics
Doctoral studentsUmberto Zannier

## Career

Bombieri published his first mathematical paper in 1957 when he was 16 years old. In 1963 at age 22 he earned his first degree (Laurea) in mathematics from the Università degli Studi di Milano under the supervision of Giovanni Ricci and then studied at Trinity College, Cambridge with Harold Davenport.

Bombieri was an assistant professor (1963–1965) and then a full professor (1965–1966) at the Università di Cagliari, at the Università di Pisa in 1966–1974, and then at the Scuola Normale Superiore di Pisa in 1974–1977. From Pisa he emigrated in 1977 to the USA, where he became a professor at the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey. In 2011 he became professor emeritus.

Bombieri is also known for his pro bono service on behalf of the mathematics profession, e.g. for serving on external review boards and for peer-reviewing extraordinarily complicated manuscripts (like the paper of Per Enflo on the invariant subspace problem).[citation needed]

## Research

The Bombieri–Vinogradov theorem is one of the major applications of the large sieve method. It improves Dirichlet's theorem on prime numbers in arithmetic progressions, by showing that by averaging over the modulus over a range, the mean error is much less than can be proved in a given case. This result can sometimes substitute for the still-unproved generalized Riemann hypothesis.

In 1969 Bombieri, De Giorgi, and Giusti solved Bernstein's problem.[10]

In 1976, Bombieri developed the technique known as the "asymptotic sieve".[11] In 1980 he supplied the completion of the proof of the uniqueness of finite groups of Ree type in characteristic 3; at the time of its publication it was one of the missing steps in the classification of finite simple groups.[12]

## Awards

Bombieri's research in number theory, algebraic geometry, and mathematical analysis have earned him many international prizes — a Fields Medal in 1974 and the Balzan Prize in 1980. He was a plenary speaker at the International Congress of Mathematicians in 1974 at Vancouver. He is a member, or foreign member, of several learned academies, including the Accademia Nazionale dei Lincei (elected 1976), the French Academy of Sciences (elected 1984), and the United States National Academy of Sciences (elected 1996).[13] In 2002 he was made Cavaliere di Gran Croce al Merito della Repubblica Italiana.[14] In 2010 he received the King Faisal International Prize (jointly with Terence Tao).[15][16] and in 2020 he was awarded the Crafoord Prize in Mathematics.[17]

## Other interests

Bombieri, accomplished also in the arts, explored for wild orchids and other plants as a hobby in the Alps when a young man.[18]

With his powder-blue shirt open at the neck, khaki pants and running shoes, he might pass for an Italian film director at Cannes. Married with a grown daughter, he is a gourmet cook and a serious painter: He carries his paints and brushes with him whenever he travels. Still, mathematics never seems far from his mind. In a recent painting, Bombieri, a one-time member of the Cambridge University chess team, depicts a giant chessboard by a lake. He's placed the pieces to reflect a critical point in the historic match in which IBM's chess-playing computers, Deep Blue, beat Garry Kasparov.[19]

## Selected publications

### Sole

• E. Bombieri, Le Grand Crible dans la Théorie Analytique des Nombres (Seconde Édition). Astérisque 18, Paris 1987.

### Joint

• Bombieri, E.; Vaaler, J. (Feb 1983). "On Siegel's lemma". Inventiones Mathematicae. 73 (1): 11–32. Bibcode:1983InMat..73...11B. doi:10.1007/BF01393823. S2CID 121274024.
• Bombieri, E.; Mueller, J. (1983). "On effective measures of irrationality for ${\displaystyle {\scriptscriptstyle {\sqrt[{r}]{a/b}}}}$  and related numbers". Journal für die Reine und Angewandte Mathematik. 342: 173–196.
• B. Beauzamy, E. Bombieri, P. Enflo and H. L. Montgomery. "Product of polynomials in many variables", Journal of Number Theory, pages 219–245, 1990.

## References

1. ^ "Site of Caccioppoli Prize".
2. ^ Premio Pitagora 2006 (in Italian)
3. ^ "Joseph L. Doob Prize".
4. ^ "2008 Doob Prize" (PDF). Notices of the AMS. 55 (4): 503–504. April 2008.
5. ^ a b "Proceedings of the International Congress of Mathematicians, 1974" (PDF). Archived from the original (PDF) on 13 November 2013.
6. ^ "Enrico Bombieri". Institute for Advanced Study. Retrieved 2019-08-07.
7. ^ a b "Enrico Bombieri PROFESSOR EMERITUS School of Mathematics Number Theory". www.ias.edu (Institute for Advanced Study). 9 December 2019. Retrieved 2 July 2021.
8. ^ Jameson, GJO. "Notes on the large sieve" (PDF). www.maths.lancs.ac.uk (University of Lancaster). Retrieved 2 July 2021.
9. ^ synopsis of text by Cambridge University Press. The Large Sieve and its Applications Arithmetic Geometry, Random Walks and Discrete Groups Part of Cambridge Tracts in Mathematics AUTHOR: E. Kowalski. Swiss Federal University (ETH), Zürich (May 2008): www.cambridge.org (Cambridge University Press). ISBN 9780521888516.
10. ^ Bombieri, Enrico; De Giorgi, Ennio; Giusti, Enrico (1969), "Minimal cones and the Bernstein problem", Inventiones Mathematicae, 7 (3): 243–268, Bibcode:1969InMat...7..243B, doi:10.1007/BF01404309, ISSN 0020-9910, MR 0250205, S2CID 59816096
11. ^ E. Bombieri, "The asymptotic sieve", Mem. Acad. Naz. dei XL, 1/2 (1976) 243–269.
12. ^ Bombieri, E. (1980). "Thompson's problem σ2=3. Appendices by A. Odlyzko and D. Hunt". Invent. Math. 58 (1): 77–100. doi:10.1007/bf01402275. S2CID 122867511. (This paper completed a line of research initiated by the Walter theorem.)
13. ^ Scheda socio Archived 2012-11-14 at the Wayback Machine, from the website of Accademia dei Lincei (elected 1976)
14. ^ Torno Armando (28 May 2002). "BOMBIERI Il re dei numeri che ha conquistato il mondo". Corriere della Sera (in Italian). p. 35.
15. ^ King Faisal Foundation, – retrieved 2010-01-11.
16. ^ "Bombieri and Tao Receive King Faisal Prize" (PDF). Notices of the AMS. 57 (5): 642–643. May 2010.
17. ^ "Crafoord Prize 2020".
18. ^ Bombieri – Mathematician retrieved 10 February 2020
19. ^ Birch, Douglas (30 September 1998). "Lifelong pursuit of mathematical pursuit Professor: At 15, Enrico Bombieri picked up a book on number theory that introduced him to the fiendishly puzzling Riemann Hypothesis. He was hooked". The Baltimore Sun.

## Sources

• Bombieri, E.; Mueller, J. (1983). "On effective measures of irrationality for ${\displaystyle {\scriptscriptstyle {\sqrt[{r}]{a/b}}}}$  and related numbers". Journal für die Reine und Angewandte Mathematik. 342: 173–196.
• Bombieri, E.; Vaaler, J. (Feb 1983). "On Siegel's lemma". Inventiones Mathematicae. 73 (1): 11–32. Bibcode:1983InMat..73...11B. doi:10.1007/BF01393823. S2CID 121274024.
• E. Bombieri, Le Grand Crible dans la Théorie Analytique des Nombres (Seconde Édition). Astérisque 18, Paris 1987.
• B. Beauzamy, E. Bombieri, P. Enflo and H. L. Montgomery. "Product of polynomials in many variables", Journal of Number Theory, pages 219–245, 1990.
• Enrico Bombieri and Walter Gubler (2006). Heights in Diophantine Geometry. Cambridge U. P.
• "Enrico Bombieri Italian mathematician". www.britannica.com (Encyclopedia Britannica). Retrieved 2 July 2021.