Manjul Bhargava

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Manjul Bhargava, Oberwolfach 2005

Manjul Bhargava (born August 8, 1974 in Hamilton (Ontario) , Canada ) is a Canadian mathematician of Indian origin who made significant contributions to number theory . He is the recipient of the Fields Medal .

Life

Bhargava was born to Indian immigrants from the Jaipur region of Canada. He grew up on Long Island in the US state of New York . His father is a chemist and his mother is a professor of mathematics at Hofstra University on Long Island. At school, on the one hand, he attracted attention because of his versatile talents, but on the other hand, he had difficulties because he simply didn't attend courses if they didn't bring him anything new. Instead, he preferred to get involved with the student magazine, play on his school's tennis and bowling teams, read books on math, and learn how to play the sitar , guitar , violin, and most importantly, tabla . By the time he was nine years old, he had already completed all of his high school math and computer science courses. He spent the second half of the tenth year with his grandparents in India. His grandfather Purushottam Lal Bhargava, a famous scholar in India, taught him Sanskrit and Indian history. He also immersed himself in playing the tabla. Upon his return, he took first place in the first New York State Science Talent Search in 1992 , which enabled him to study mathematics at Harvard University from 1992.

After only one year, Bhargava was awarded the Detur Prize of Harvard University for his outstanding academic achievements. At the age of 19 he was appointed as a teaching fellow. For his success in teaching he received the Derek Bok Award from Harvard University three times from 1993 to 1995. In 1996 he graduated from Harvard University with a Bachelor of Arts and a summa cum laude grade . He was honored for his excellent work by being awarded the Hoopes Prize from Harvard University.

From 1996 Bhargava studied mathematics at Princeton University , where he received his doctorate summa cum laude with Andrew Wiles in 2001 with a thesis on higher composition laws . Two years later (still at the age of 28) he received a professorship for mathematics at Princeton University (full professor with tenure). He is one of the youngest scientists ever to be appointed to such a professorship at Princeton.

In addition to mathematics, playing the tabla is one of Bhargava's great passions. He was tutored several times by Zakir Hussain , one of the most famous and high-profile tabla players of our time. Bhargava occasionally appears at public concerts at Harvard and Princeton. In 2003 he took part in the GigaPop Ritual , a distributed live concert for digital dholaks , electronic didgeridoos , electronic violins, rbow, sitar, tabla and bass guitar played on site by musicians and scientists from McGill University and Princeton University.

He was one of the math advisors in the film adaptation of the life of S. Ramanujan The Poetry of Infinite (and associate producer of the film).

Services

During his studies at Harvard, Bhargava wrote four original papers in which he standardized and generalized several results of famous mathematicians and solved several known problems. In these works there was a new generalization of the faculty . He applied this new faculty function to rings of integer polynomials and p-adic analysis, among other things.

He suddenly became famous for his dissertation in which he added higher compositional laws to the Gaussian composition of binary integer square forms known since 1801 ( Disquisitiones Arithmeticae ) , for example for binary integer cubic forms. He found a general theory into which these laws fit, and found that there are at least 14 such higher number theoretic laws of composition, one of which is Gauss's for binary quadratic forms. These groundbreaking and completely surprising results were published in a series of four papers in the Annals of Mathematics from 2004 to 2008 . He published on the asymptotic density of the discriminants of quartic and quintic number fields. He also achieved results in the field of Cohen - Martinet heuristics for class numbers of algebraic number fields (similar to the Cohen- Lenstra heuristics for class numbers of square number fields).

His simple proof of the so-called 15 theorem by John Horton Conway and William Schneeberger also received a lot of attention . In the meantime, together with Jonathan P. Hanke, he has also proven Conway's conjecture known as the 290 theorem . Conway's 15 theorem says that if an integer square form with an integer matrix (that is, not only the coefficients are integers, but the non-diagonal elements are also even) represents the natural numbers up to 15, it represents all natural numbers. The complicated 1993 proof by Conway and WA Schneeberger was never published. Bhargava found a simpler proof. A similar theorem applies to integer square shapes without limitation, replacing 15 with 290. It was also proven by Bhargava with Jonathan Hanke.

With Arul Shankar he proved the boundedness of the mean rank of elliptic curves over the rational numbers and they proved that a positive measure of the elliptic curves over the rational numbers has rank 0 and satisfies the conjecture of Birch and Swinnerton-Dyer , one of the Millennium problems . In 2014, Bhargava, Christopher Skinner, and Wei Zhang showed that this is the case for the majority (over 66 percent) of elliptical curves. He gave simpler proofs and new interpretations for the theorems of Harold Davenport and Hans Heilbronn about the density of discriminants of cubic number fields . In 2013, he proved that most of the hyperelliptic curves defined over the rational numbers have no rational points.

Honors

In 1996 Bhargava was awarded the Frank and Brennie Morgan Prize, jointly awarded by the American Mathematical Society , the Mathematical Association of America and the Society for Industrial and Applied Mathematics , for work written during his studies at Harvard . In 2003 he received the Merten M. Hasse Prize of the Mathematical Association of America and in 2004 the Leonard M. and Eleanor B. Blumenthal Award . In 2005 he was honored with the Clay Research Award and the SASTRA Ramanujan Prize . In 2008 he received the prestigious Frank Nelson Cole Prize for Number Theory from the American Mathematical Society. In 2011 he was awarded the Fermat Prize and elected to the National Academy of Sciences in 2013. He is a Fellow of the American Mathematical Society and of the American Academy of Arts and Sciences since 2017 and the Royal Society since 2019 .

In 2006 he was invited speaker at the International Congress of Mathematicians in Madrid ( Higher composition laws and applications ). He was selected as plenary speaker at the International Congress of Mathematicians 2014 in Seoul (Rational points on elliptic and hyperelliptic curves) and received the Fields Medal there . In 2015 he was awarded the Padma Bhushan .

Fonts

Web links

Individual evidence

  1. ^ Bhargava P-orderings and polynomial functions on arbitrary subsets of Dedekind rings , Journal für Reine und Angewandte Mathematik 490 (1997) 101-127
  2. ↑ Bhargava Generalized factorials and fixed divisors over subsets of a Dedekind domain , J. Number Theory 72 (1998) 67-75
  3. ↑ Bhargava The Factorial Function and Generalizations , American Mathematical Monthly, Volume 107, 2000, pp. 783-799
  4. ↑ Bhargava On P-orderings, rings of integer-valued polynomials, and ultrametric analysis , Journal of the AMS, 22, 2009, 963-993
  5. ↑ Bhargava Higher composition laws , part 1-4, Annals of Mathematics, Volume 159, 2004, pp. 217-250, 865-886, 1329-1360, Volume 167, 2008, pp. 53-94, Part 1: A new view on Gauss composition, and quadratic generalizations, Part 2: On cubic analogues of Gauss composition, Part 3: The parametrization of quartic rings, Part 4: The parametrization of quintic rings
  6. ↑ Bhargava The density of discriminants of quintic rings and fields The Annals of Mathematics, Volume 172, 2010, pp. 1559-1591
  7. ↑ Bhargava The density of discriminants of quartic rings and fields , Annals of Mathematics, Volume 162, 2005, pp. 1031-1063
  8. ↑ On Bhargava´s proof and the history of the problem: John Horton Conway Universal quadratic forms and the Fifteen Theorem , Contemporary Mathematics, pdf
  9. ↑ Bhargava On the Conway-Schneeberger fifteen theorem , in Quadratic forms and their applications (Dublin 1999) , Contemporary Mathematics 272, American Mathematical Society 2000, pp. 27-37
  10. ↑ Bhargava, Shankar Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves , 2010
  11. ↑ Bhargava, Shankar Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0 , 2010, Arxiv
  12. ↑ Bhargava, Skinner, Zhang, A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture , Arxiv 2014
  13. ^ Bhargava, Arul Shankar, Jacob Tsimerman On the Davenport-Heilbronn theorems and second order terms
  14. ↑ A hyperelliptic curve is defined by an equation , where f is a polynomial of degree with n different zeros. It is defined over the rational numbers if f has rational coefficients.
personal data
SURNAME Bhargava, Manjul
BRIEF DESCRIPTION Canadian mathematician
DATE OF BIRTH August 8, 1974
PLACE OF BIRTH Hamilton, Ontario , Canada