Life

Born in New York City

New York City

New York is the most populous city in the United States and the center of the New York Metropolitan Area, one of the most populous metropolitan areas in the world. New York exerts a significant impact upon global commerce, finance, media, art, fashion, research, technology, education, and...

, Mazur attended the Bronx High School of Science

Bronx High School of Science

The Bronx High School of Science is a specialized New York City public high school often considered the premier science magnet school in the United States. Founded in 1938, it is now located in the Bedford Park section of the Bronx...

 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

Princeton University

Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....

 in 1959, becoming a Junior Fellow at Harvard from 1961 to 1964. He is currently the Gerhard Gade University Professor and a Senior Fellow at Harvard. In 1982 he was elected a member of the National Academy of Sciences

United States National Academy of Sciences

The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...

. Mazur has received the Veblen Prize in geometry, the Cole Prize

Cole Prize

The Frank Nelson Cole Prize, or Cole Prize for short, is one of two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory. The prize is named after Frank Nelson Cole, who...

 in number theory, the Chauvenet Prize for exposition, and the Steele Prize for seminal contribution to research from the American Mathematical Society

American Mathematical Society

The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...

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Work

His early work was in geometric topology

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...

. In a clever, elementary fashion, he proved the generalized Schoenflies conjecture (his complete proof required an additional result by Marston Morse

Marston Morse

Harold Calvin Marston Morse was an American mathematician best known for his work on the calculus of variations in the large, a subject where he introduced the technique of differential topology now known as Morse theory...

), around the same time as Morton Brown

Morton Brown

Morton Brown is an American mathematician, who specializes in geometric topology.In 1958 Brown earned his PhD from the University of Wisconsin at Madison under R. H. Bing. From 1960 to 1962 he was at the Institute for Advanced Study...

. Both Brown and Mazur received the Veblen Prize for this achievement. He also discovered the Mazur manifold

Mazur manifold

In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth 4-dimensional manifold which is not diffeomorphic to the standard 4-ball...

 and the Mazur swindle.

His observations in the 1960s on analogies between primes and knots

Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...

 were taken up by others in the 1990s giving rise to the field of Arithmetic topology

Arithmetic topology

Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. In the 1960s topological interpretations of class field theory were given by John Tate based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier based on Étale cohomology...

.

Coming under the influence of Alexander Grothendieck

Alexander Grothendieck

Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...

's approach to algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, he moved into areas of diophantine geometry

Diophantine geometry

In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...

. Mazur's torsion theorem

Mazur's torsion theorem

In algebraic geometry Mazur's torsion theorem, due to Barry Mazur, classifies the possible torsion subgroups of the group of rational points on an elliptic curve defined over the rational numbers....

, which gives a complete list of the possible torsion subgroups of elliptic curve

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

s 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 curve

Modular curve

In number theory and algebraic geometry, a modular curve Y is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL...

s. 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 Andrew Wiles

Andrew Wiles

Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory...

's ultimately successful attack on Fermat's last theorem

Fermat's Last Theorem

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....

. Mazur and Wiles had earlier worked together on the main conjecture of Iwasawa theory

Iwasawa theory

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur...

.

In an expository paper, Number Theory as Gadfly

Gadfly (social)

A gadfly is a person who upsets the status quo by posing upsetting or novel questions, or just being an irritant. The term has been used to describe many politicians and social commentators....

, 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

Imagining Numbers

Mathematician Barry Mazur authored a book entitled Imagining Numbers: . Farrar, Straus and Giroux published the book in 2003...

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External links

• Homepage of Barry Mazur