Barry Charles Mazur | |
|---|---|
 Mazur in 1992 | |
| Born | (1937-12-19)December 19, 1937 (age 87) New York City, U.S. |
| Education | Massachusetts Institute of Technology Princeton University (PhD) |
| Known for | Diophantine geometry Generalized Schoenflies conjecture Artin–Mazur zeta function Eilenberg–Mazur swindle Fontaine–Mazur conjecture Mazur manifold Mazur's Conjecture B Mazur's control theorem Mazur's torsion theorem |
| Awards | Chern Medal (2022) National Medal of Science (2011) Chauvenet Prize (1994) Cole Prize (1982) Veblen Prize (1966) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Harvard University |
| Doctoral advisor | Ralph Fox R. H. Bing |
| Doctoral students | |
Barry Charles Mazur (/ˈmeɪzər/;[1] born December 19, 1937) is an American mathematician and theGerhard Gade University Professor atHarvard University.[2] His contributions to mathematics include his contributions toWiles's proof of Fermat's Last Theorem innumber theory,Mazur's torsion theorem inarithmetic geometry, theMazur swindle ingeometric topology, and theMazur manifold indifferential topology.
Born in New York City, Mazur attended theBronx High School of Science, and left after his junior year to attendMIT;[3] he did not graduate from the university on account of failing a then-presentROTC requirement. He was nonetheless accepted for graduate studies atPrinceton University, where he received his PhD in mathematics in 1959 after completing a doctoral dissertation titledOn embeddings of spheres.[4] Thus, his only academic degree is a PhD.[3] He then became a Junior Fellow atHarvard University from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard. He is the brother ofJoseph Mazur and the father ofAlexander J. Mazur.[5]
His early work was in geometric topology. In an elementary fashion, he proved thegeneralized Schoenflies conjecture (his complete proof required an additional result byMarston Morse), around the same time asMorton Brown. Both Brown and Mazur received theVeblen Prize for this achievement. He also discovered theMazur manifold and theMazur swindle.
His observations in the 1960s on analogies betweenprimes andknots were taken up by others in the 1990s giving rise to the field ofarithmetic topology.
Coming under the influence ofAlexander Grothendieck's approach toalgebraic geometry, he moved into areas ofdiophantine geometry.Mazur's torsion theorem, which gives a complete list of the possible torsion subgroups ofelliptic 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 certainmodular 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 inWiles's proof of Fermat's Last Theorem. Mazur and Wiles had earlier worked together on themain conjecture of Iwasawa theory.
In an expository paper,Number Theory asGadfly,[6] 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 bookImagining Numbers[7] andCircles Disturbed, a collection of essays on mathematics and narrative that he edited with writerApostolos Doxiadis.[2]
Mazur was elected to theAmerican Academy of Arts and Sciences in 1978.[8] In 1982 he was elected a member of theNational Academy of Sciences.[9] Mazur was elected to theAmerican Philosophical Society in 2001,[10] and in 2012 he became a fellow of theAmerican Mathematical Society.[11]
Mazur has received theVeblen Prize in geometry (1966), theCole Prize in number theory (1982), the Chauvenet Prize for exposition (1994),[6] and theSteele Prize for seminal contribution to research (2000) from theAmerican Mathematical Society. In early 2013, he was presented with one of the 2011National Medals of Science by PresidentBarack Obama.[12] In 2022, he was awarded theChern Medal for outstanding lifelong achievement in mathematics.[13]
