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 university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...

. 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 Sir 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 ...

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, 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 in

geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...

, 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 is a State school, public Specialized high schools in New York City, 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 ...

, and left after his junior year to attend

MIT The Massachusetts Institute of Technology (MIT) is a private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of modern technology and sc ...

; he did not graduate from the university 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. While ROTC graduate officers serve in all branches o ...

requirement. He was nonetheless accepted for graduate studies at

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

, where he received his PhD in mathematics in 1959 after completing a doctoral dissertation titled ''On embeddings of spheres''. Thus, his only academic degree is a PhD. 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 and the father of Alexander J. Mazur.

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 Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) 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 a ...

), 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 A knot is a fastening in rope or interwoven lines. Knot or knots may also refer to: Other common meanings * Knot (unit), of speed * Knot (wood), a timber imperfection Arts, entertainment, and media Films * ''Knots'' (film), a 2004 film * ''Kn ...

were taken up by others in the 1990s giving rise to the field of arithmetic topology. Coming under the influence of

Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...

's approach to

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, he moved into areas of

diophantine geometry In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...

. Mazur's torsion theorem, which gives a complete list of the possible torsion subgroups of

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

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 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 g ...

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 mathematics, the main conjecture of Iwasawa theory is a deep relationship between ''p''-adic ''L''-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and ...

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

Mazur was elected to the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) 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, and other ...

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, NGO, 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 ...

. Mazur was elected to the

American Philosophical Society The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...

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 numbe ...

in number theory (1982), the Chauvenet Prize for exposition (1994), and the

Steele Prize The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories. The prizes have b ...

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 was the 44th president of the United States from 2009 to 2017. A member of the Democratic Party, he was the first African American president in American history. O ...

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

Publications

Books

* * * *

References

External links

Barry Mazur
Harvard University *
Video of Mazur talking about his work
from the National Science & Technology Medals Foundation
Barry Mazur and the Infinite Cheese of Knowledge
a documentary about him and his family, made by Oliver Ralfe. {{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 American number theorists The Bronx High School of Science alumni Princeton University alumni Harvard University Department of Mathematics faculty National Medal of Science laureates Institute for Advanced Study visiting scholars Fellows of the American Mathematical Society American topologists Mathematicians from New York (state) 20th-century American Jews 21st-century American Jews Jewish American scientists Members of the American Philosophical Society Massachusetts Institute of Technology alumni