Barry Mazur

Barry Charles Mazur
Barry Mazur in 1992
Born	(1937-12-19) December 19, 1937 New York City, New York
Nationality	American
Fields	Mathematics
Institutions	Harvard University
Alma mater	Princeton University
Doctoral advisor	Ralph Fox R. H. Bing
Doctoral students	Nigel Boston Noam Elkies Jordan Ellenberg Minhyong Kim David Goss Michael Harris Daniel Kane Michael McQuillan Victor S. Miller Paul Vojta
Known for	diophantine geometry generalized Schoenflies conjecture Mazur swindle Mazur's torsion theorem
Notable awards	National Medal of Science (2011) Chauvenet Prize (1994) Cole Prize (1982) Veblen Prize (1966)

Barry Charles Mazur (born December 19, 1937) is an American mathematician and a Gerhard Gade University Professor at Harvard University.

Life

Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph.D. from Princeton University in 1959, becoming a Junior Fellow at Harvard from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard.

Work

His early work was in geometric topology. In an elementary fashion, he proved the generalized Schoenflies conjecture (his complete proof required an additional result by Marston Morse), around the same time as Morton Brown. Both Brown and Mazur received the Veblen Prize for this achievement. He also discovered the Mazur manifold and the Mazur swindle.

His observations in the 1960s on analogies between primes and knots were taken up by others in the 1990s giving rise to the field of arithmetic topology.

Coming under the influence of Alexander Grothendieck's approach to algebraic geometry, he moved into areas of diophantine geometry. Mazur's torsion theorem, which gives a complete list of the possible torsion subgroups of elliptic curves over the rational numbers, is a deep and important result in the arithmetic of elliptic curves. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves. This proof was carried in his seminal paper "Modular curves and the Eisenstein ideal". The ideas of this paper and Mazur's notion of Galois deformations, were among the key ingredients in Wiles's proof of Fermat's Last Theorem. Mazur and Wiles had earlier worked together on the main conjecture of Iwasawa theory.

In an expository paper, Number Theory as Gadfly, Mazur describes number theory as a field which

produces, without effort, innumerable problems which have a sweet, innocent air about them, tempting flowers; and yet... number theory swarms with bugs, waiting to bite the tempted flower-lovers who, once bitten, are inspired to excesses of effort!

He expanded his thoughts in the 2003 book Imagining Numbers and Circles Disturbed, a collection of essays on mathematics and narrative that he edited with writer Apostolos Doxiadis.

Awards and honors

In 1982 he was elected a member of the National Academy of Sciences, and in 2012 he became a fellow of the American Mathematical Society.

Mazur has received the Veblen Prize in geometry, the Cole Prize in number theory, the Chauvenet Prize for exposition, and the Steele Prize for seminal contribution to research from the American Mathematical Society. In early 2013, he was presented with one of the 2011 National Medals of Science by President Barack Obama.

See also

• Fontaine–Mazur conjecture
• Mazur's control theorem

External links

• Homepage of Barry Mazur
• .
• Video of Mazur talking about his work, from the National Science & Technology Medals Foundation