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, "Mat ...

Mazur's torsion theorem In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in term ...

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, and the Mazur manifold in differential topology.

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 The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...

, 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 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 ...

, 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 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 Morton Brown (born August 12, 1931, in New York City, New York) is an American mathematician, who specializes in geometric topology. In 1958 Brown earned his Ph.D. from the University of Wisconsin-Madison under R. H. Bing. From 1960 to 1962 he w ...

. Both Brown and Mazur received the

Veblen Prize __NOTOC__ The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was founded in 1961 in memory of Oswald Veblen. The Veblen Prize is now worth US$5000, and is ...

for this achievement. He also discovered the Mazur manifold and the Mazur swindle. His observations in the 1960s on analogies between

primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

and

knots A knot is a fastening in rope or interwoven lines. Knot may also refer to: Places * Knot, Nancowry, a village in India Archaeology * Knot of Isis (tyet), symbol of welfare/life. * Minoan snake goddess figurines#Sacral knot Arts, entertainme ...

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

arithmetic topology Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds. Analogies The following are some of the analogies used ...

. Coming under the influence of Alexander Grothendieck's approach to algebraic geometry, 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 ...

, 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 ...

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 ...

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 Apostolos K. Doxiadis ( el, Απόστολος Κ. Δοξιάδης; born 1953) is a Greek writer. He is best known for his international bestsellers '' Uncle Petros and Goldbach's Conjecture'' (2000) and ''Logicomix'' (2009). Early life Doxiad ...

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, a ...

in 1978. In 1982 he was elected a member of the National Academy of Sciences. 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 communit ...

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

in geometry (1966), the Cole Prize 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, Obama was the first African-American president of the ...

. In 2022, he was awarded the

Chern Medal The Chern Medal is an international award recognizing outstanding lifelong achievement of the highest level in the field of mathematics. The prize is given at the International Congress of Mathematicians (ICM), which is held every four years. ...

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