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

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...

. His contributions to mathematics include his contributions to

Wiles's proof of Fermat's Last Theorem Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Ferma ...

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, Mazur's torsion theorem in

arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...

, the

Mazur swindle Mazur can refer to: * Masurians or Mazurs, an ethnic group with historic origins in the Polish region of Masovia * Mazur (surname), including a list of people so named * Mazur (dance), a traditional Polish folk dance * Mazur, Iran, a village in Mar ...

geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originat ...

, and the Mazur manifold in

differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

Life

Born in New York City, Mazur attended the

Bronx High School of Science The Bronx High School of Science, commonly called Bronx Science, is a public specialized high school in The Bronx in New York City. It is operated by the New York City Department of Education. Admission to Bronx Science involves passing the Sp ...

and MIT, although he did not graduate from the latter on account of failing a then-present

ROTC The Reserve Officers' Training Corps (ROTC ( or )) is a group of college- and university-based officer-training programs for training commissioned officers of the United States Armed Forces. Overview While ROTC graduate officers serve in all ...

requirement. He was nonetheless accepted for graduate studies at

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...

, 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

from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard. He is the brother of

Joseph Mazur Joseph C. Mazur (born in the Bronx in 1942) is Professor Emeritus of Mathematics at Marlboro College, in Marlboro, Vermont. He holds a B.S. from Pratt Institute, where he first studied architecture. He spent his junior year in Paris, studyi ...

and the father of

Alexander J. Mazur Alexander J. Mazur (b. 1969, d. August 2016; also known as Zeke Mazur) was an American scholar specializing in Neoplatonism, Gnosticism, and mysticism. Mazur is noted for his work on the relationship between Neoplatonism and Gnosticism. Educ ...

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

. 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 curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) 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 ...

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

. Mazur and Wiles had earlier worked together on the main conjecture of Iwasawa theory. In an expository paper, ''Number Theory as

Gadfly Gadfly most commonly refers to: * Horse-fly or Botfly * Gadfly (philosophy and social science), a person who upsets the status quo Gadfly may also refer to: Entertainment * ''The Gadfly'', an 1897 novel by Ethel Lilian Voynich ** ''The Gadfly'' ...

'', Mazur describes number theory as a field which He expanded his thoughts in the 2003 book ''

Imagining Numbers ''Imagining Numbers: (particularly the square root of minus fifteen)'' is a 2003 popular mathematics book by mathematician Barry Mazur The aim of the book is not a history of imaginary numbers but an attempt to :''re-create, in ourselves, the s ...

'' and ''Circles Disturbed, a collection of essays on mathematics and narrative'' that he edited with writer Apostolos Doxiadis.

Awards and honors

Mazur was elected to the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, ...

in 1978. In 1982 he was elected a member of the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nat ...

. Mazur was elected to the

American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communi ...

in 2001, and in 2012 he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...

. Mazur has received the Veblen Prize in geometry (1966), the

Cole Prize The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-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 ...

in number theory (1982), the Chauvenet Prize for exposition (1994), and the Steele Prize for seminal contribution to research (2000) from the

. In early 2013, he was presented with one of the 2011 National Medals of Science by President

Barack Obama Barack Hussein Obama II ( ; born August 4, 1961) is an American politician who served as the 44th president of the United States from 2009 to 2017. A member of the Democratic Party (United States), Democratic Party, Obama was the first Af ...

. In 2022, he was awarded the Chern Medal for outstanding lifelong achievement in mathematics.

Publications

Books

* * * *

References

External links

Homepage of Barry Mazur
*
Video of Mazur talking about his work
from the National Science & Technology Medals Foundation
Barry Mazur on MathSciNet
{{DEFAULTSORT:Mazur, Barry 1937 births Members of the United States National Academy of Sciences Living people 20th-century American mathematicians 21st-century American mathematicians Mathematics popularizers Number theorists 20th-century American Jews The Bronx High School of Science alumni Princeton University alumni Harvard University faculty National Medal of Science laureates Institute for Advanced Study visiting scholars Fellows of the American Mathematical Society Topologists Mathematicians from New York (state) 21st-century American Jews Members of the American Philosophical Society