Barry Mazur

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Barry Mazur

Barry Charles Mazur (born December 19, 1937 in New York City ) is an American mathematician who works in the fields of topology , number theory and arithmetic algebraic geometry .

life and work

Mazur attended the Bronx High School of Science and then studied at the Massachusetts Institute of Technology (MIT) and at Princeton University , where he received his PhD in 1959 under Ralph Fox and RH Bing ( On the embedding of spheres ), while at the same time he was a Research Fellow from 1958 at the neighboring Institute for Advanced Study . He then went to Harvard , where he first became a Junior Fellow , becoming an assistant professor in 1962, associate professor in 1965 and professor of mathematics from 1969 (from 1982 William Petschek Professor , from 1998 Gerhard Gade Professor ). He is a regular guest resident at IHES near Paris.

Mazur initially worked in the field of geometric topology, where he simultaneously with Morton Brown proved the generalized Schoenflies conjecture , which says in descriptive terms that a two-dimensional knotted sphere in three-dimensional space (as opposed to a "thread") always becomes one normal sphere can be untied - without the use of cuts or "perforations" (with an analogous assumption in higher dimensions). Together with Brown he received the Oswald Veblen Prize for this .

As a student of Alexander Grothendieck , who lectured regularly at Harvard in the early 1960s, he then shifted his interest to algebraic geometry, which was followed by number theory.

In On Modular curves and the Eisenstein Ideal from 1978 he classified the possible torsional subgroups (finite subgroups) of the group of rational points on elliptic curves. Let Z n be the cyclic group of order n , then the possible torsion subsets of elliptic curves over the rational numbers Z n with 1 ≤  n  ≤ 10, as well as Z 12 ; and the direct sum of Z 2 with Z 2 , Z 4 , Z 6 or Z 8 .

In Class fields of abelian extensions of Q ( Inventiones Mathematicae 1984) Mazur and Andrew Wiles proved the main conjecture of Iwasawa theory (a branch of algebraic number theory founded by Kenkichi Iwasawa ). Mazur also originated the concept of the deformation of Galois representations , which played an important role in Andrew Wiles' proof of the Taniyama-Shimura conjecture and the great Fermat theorem.

His PhD students include Noam Elkies , Glenn Stevens , Michael McQuillan , Matthew Emerton , Michael Harris , Daniel Kubert , Cathy O'Neil and Paul Vojta .

He has been married to the writer Grace Dane since 1960 and has one child.

Awards and honors (selection)

In 1983 he gave a plenary lecture at the ICM in Warsaw ( Modular Curves and Arithmetic ) and in 1974 he was invited speaker at the ICM in Vancouver ( P-adic analytic number theory of elliptic curves and abelian varieties over Q ). In 2002 he was invited speaker at the International Congress of Mathematicians in Beijing with Karl Rubin (Elliptic curves and class field theory).

Fonts

  • Barry Mazur: Imagining numbers: particularly the square root of minus fifteen . Farrar Straus Giroux, 2004, ISBN 0-374-17469-5 (popular book).
  • B. Mazur Arithmetic on Curves In: Bulletin American Mathematical Society. 1986, p. 207 (especially on Falting's theorem, ams.org ).
  • B. Mazur: Number theory as gadfly . In: American Mathematical Monthly . tape 98 , 1991, pp. 593–610 , doi : 10.2307 / 2324924 (on the background of Wiles' proof of the Shimura-Taniyama conjecture, Mazur received the Chauvenet Prize for it).
  • Barry Mazur, Michael Artin : Etale Homotopy . Springer, 1969, ISBN 3-540-04619-4 .
  • Morris W. Hirsch , Barry Mazur: Smoothings of Piecewise Linear Manifolds . Princeton Univ. Pr, 1974, ISBN 0-691-08145-X .
  • Nicholas M. Katz , Barry Mazur: Arithmetic Moduli of Elliptic Curves . Princeton Univ. Pr, 1985, ISBN 0-691-08349-5 .
  • B. Mazur: Perturbations, deformations, and variations (and "near-misses") in geometry, physics, and number theory . In: Bull. Amer. Math. Soc. tape 41 , no. 3 , 2004, p. 307-336 ( ams.org [accessed April 9, 2011]).
  • B. Mazur: Deformation of Galois representations. In: Cornell, Silverman, Stevens * Barry Mazur: Modular forms and Fermats last theorem. Springer 1997.
  • B. Mazur: Deforming Galois representations. In: Ihara, Ribet, Serre (Eds.): Galois groups over Q. MSRI Publications Volume 16, Springer 1989.
  • B. Mazur, Peter Swinnerton-Dyer : Arithmetic of Weil curves . In: Inventiones Mathematicae . tape 25 , no. 1 , 1974, p. 1–61 , doi : 10.1007 / BF01389997 ( gdz.sub.uni-goettingen.de [accessed on April 9, 2011]).
  • B. Mazur, Dorian Goldfeld : Rational isogenies of prime degree . In: Inventiones Mathematicae . tape 44 , no. 2 , 1978, p. 129–162 , doi : 10.1007 / BF01390348 ( gdz.sub.uni-goettingen.de [accessed on April 9, 2011]).
  • B. Mazur, Andrew Wiles : Class fields of abelian extensions of Q . In: Inventiones Mathematicae . tape 76 , no. 2 , 1984, p. 179-330 , doi : 10.1007 / BF01388599 ( gdz.sub.uni-goettingen.de [accessed April 9, 2011]).
  • B. Mazur, John T. Tate , Jeremy Teitelbaum : On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer . In: Inventiones Mathematicae . tape 84 , no. 1 , 1986, pp. 1-48 , doi : 10.1007 / BF01388731 ( gdz.sub.uni-goettingen.de [accessed on April 9, 2011]).
  • with Robert F. Coleman : The eigencurve , in: Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., 254, Cambridge University Press 1998, pp. 1-113.
  • B. Mazur: Finding meaning in error terms . In: Bull. Amer. Math. Soc. tape 45 , no. 2 , 2008, p. 185-228 ( ams.org [accessed April 9, 2011]).
  • with Eric Friedlander Filtrations on the homology of algebraic varieties , American Mathematical Society 1994
  • with William A. Stein : Prime numbers and the Riemann Hypothesis , Cambridge UP 2015 ( website for the book with preprint)
  • The unity and breadth of mathematics - from Diophantus to today , Paul Bernays Lecture, ETH Zurich 2018, pdf

Web links

Remarks

  1. ↑ The change was due to the fact that MIT at that time required participation in reserve officer courses, which Mazur only completed listlessly and was therefore not recognized.
  2. ↑ Barry Mazur: On the embedding of spheres. In: Bull. Amer. Math. Soc. 65, 1959, pp. 59-65, ( doi: 10.1090 / S0002-9904-1959-10274-3 ).
personal data
SURNAME Mazur, Barry
ALTERNATIVE NAMES Mazur, Barry Charles (full name)
BRIEF DESCRIPTION American mathematician
DATE OF BIRTH December 19, 1937
PLACE OF BIRTH New York City