Barry Mazur

File:Barry Mazur - National Medal of Science, 2011.webm

Born in New York City, Mazur attended the Bronx High School of Science, and left after his junior year to attend MIT;{{Cite book |last=Krantz |first=Steven G. |url=https://books.google.com/books?id=8mBdvAjk_gQC&q=Mazur |title=Mathematical Apocrypha Redux |date=2005 |publisher=The Mathematical Association of America |isbn=0-88385-554-2 |page=38 |language=en}} he did not graduate from the university on account of failing a then-present ROTC requirement. He was nonetheless accepted for graduate studies at Princeton University, where he received his PhD in mathematics in 1959 after completing a doctoral dissertation titled On embeddings of spheres.{{Cite book|last=Mazur|first=Barry Charles|url=https://catalog.princeton.edu/catalog/2843260|title=On embeddings of spheres|date=1959|language=en}} Thus, his only academic degree is a PhD. 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.He talks about his son about minute 24 of the documentary [https://www.youtube.com/watch?v=bEf26NFur3w Barry Mazur and the Infinite Cheese of Knowledge].

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". The ideas of this paper and Mazur's notion of Galois deformations were among the key ingredients in Wiles's proof of Fermat's Last Theorem. Mazur and Wiles had earlier worked together on the main conjecture of Iwasawa theory.

In an expository paper, Number Theory as Gadfly,{{cite journal|author=Mazur, Barry|title=Number Theory as Gadfly|journal= Amer. Math. Monthly|volume=98|year=1991|issue=7|pages=593–610|url=http://www.maa.org/programs/maa-awards/writing-awards/number-theory-as-gadfly|doi=10.2307/2324924|jstor=2324924}} Mazur describes number theory as a field which

{{blockquote|"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{{cite book |author=Mazur, Barry |title=Imagining numbers: (particularly the square root of minus fifteen) |publisher=Penguin Books |location=New York |year=2004 |isbn=0-14-100887-3 |url=https://books.google.com/books?id=i3ik02nXA0wC}} and Circles Disturbed, a collection of essays on mathematics and narrative that he edited with writer Apostolos Doxiadis.

Mazur was elected to the American Academy of Arts and Sciences in 1978.{{Cite web|title=Barry Charles Mazur|url=https://www.amacad.org/person/barry-charles-mazur|access-date=October 15, 2021|website=American Academy of Arts & Sciences|language=en}} In 1982 he was elected a member of the National Academy of Sciences.{{Cite web|title=Barry C. Mazur|url=http://www.nasonline.org/member-directory/members/52830.html|access-date=October 15, 2021|website=www.nasonline.org}} Mazur was elected to the American Philosophical Society in 2001,{{Cite web|title=APS Member History|url=https://search.amphilsoc.org/memhist/search?creator=Barry+Mazur&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced|access-date=October 15, 2021|website=search.amphilsoc.org}} and in 2012 he became a fellow of the American Mathematical Society.[https://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved February 4, 2013.

Mazur has received the Veblen Prize 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 American Mathematical Society. In early 2013, he was presented with one of the 2011 National Medals of Science by President Barack Obama.{{cite web|url=https://www.bostonglobe.com/metro/2013/01/07/local-professors-honored-white-house-ceremony/tMHwucDtwrPcZvUXO9bM0I/story.html|title=3 local professors to get US honors|date=January 7, 2013|publisher=Boston Globe|access-date=October 24, 2016}} In 2022, he was awarded the Chern Medal for outstanding lifelong achievement in mathematics.{{cite web|url=https://www.mathunion.org/imu-awards/chern-medal-award/chern-medal-award-2022|title=Chern Medal Award 2022|access-date=July 5, 2022}}

• {{cite book |last1=Mazur|first1=Barry|title=Prime numbers and the Riemann hypothesis|publisher=Cambridge University Press|location=New York, NY|year=2016|isbn=978-1-107-49943-0|last2=Stein|first2=William}}
• {{cite book |title=Collected works of John Tate : parts i and ii|publisher=American Mathematical Society|year=2016|isbn=978-0-8218-9091-2|editor-first=Barry|editor-last=Mazur|editor-first2=Serre|editor-last2=Jean-Pierre}}
• {{cite book |last=Mazur|first=Barry|title=Imagining numbers : (particularly the square root of minus fifteen)|publisher=Farrar Straus Giroux|location=New York|year=2003|isbn=0-312-42187-7|mr=1950850}}
• {{cite book|last1=Katz|first1=Nicholas M.|last2=Mazur|first2=Barry|title=Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, 108|date=1985|publisher=Princeton University Press|isbn=0-691-08349-5|mr=0772569}}

• Eigencurve
• Eisenstein ideal
• Iwasawa theory
• Microbundle
• Tate vector space

{{reflist}}

{{wikiquote}}

• [https://sites.harvard.edu/barry-mazur/ Barry Mazur], Harvard University
• {{MacTutor Biography|id=Mazur_Barry}}
• [https://www.youtube.com/watch?v=b09VmDM5lJQ Video of Mazur talking about his work], from the National Science & Technology Medals Foundation
• [https://www.youtube.com/watch?v=bEf26NFur3w Barry Mazur and the Infinite Cheese of Knowledge], a documentary about him and his family, made by Oliver Ralfe.

{{Winners of the National Medal of Science|physical}}

{{Chauvenet Prize recipients}}

{{Veblen Prize recipients}}

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Category:1937 births

Category:Members of the United States National Academy of Sciences

Category:Living people

Category:20th-century American mathematicians

Category:21st-century American mathematicians

Category:Mathematics popularizers

Category:American number theorists

Category:The Bronx High School of Science alumni

Category:Princeton University alumni

Category:Harvard University Department of Mathematics faculty

Category:National Medal of Science laureates

Category:Institute for Advanced Study visiting scholars

Category:Fellows of the American Mathematical Society

Category:American topologists

Category:Mathematicians from New York (state)

Category:20th-century American Jews

Category:21st-century American Jews

Category:Jewish American scientists

Category:Members of the American Philosophical Society

Category:Massachusetts Institute of Technology alumni