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Barry Mazur
Summary
Barry Charles Mazur (ˈmeɪzʊr; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.
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. He was nonetheless accepted for graduate studies at Princeton University, from where he received his PhD in mathematics in 1959 after completing a doctoral dissertation titled "On embeddings of spheres." He then became a Junior Fellow at Harvard University from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard. He is the brother of Joseph Mazur and the father of Alexander J. Mazur.
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".
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