Mathematics For All

Grigory Margulis (born February 24, 1946) - The Fields Medalist of 1978

Gregori Margulis began his undergraduate studies at Moscow University in 1962, earning his first degree in 1967 and continuing for postgraduate studies. He demonstrated significant mathematical potential, winning the young mathematicians prize from the Moscow Mathematical Society in 1968 during his postgraduate tenure. Margulis completed his graduate studies in 1970, receiving the degree of Candidate of Science for his thesis "On some problems in the theory of U-systems."

Margulis, after earning the Candidate of Science degree, worked at the Institute for Problems in Information Transmission as a Junior scientific worker from 1970 to 1974, then became a Senior scientific worker. He was awarded a Fields Medal at the International Congress in Helsinki but could not attend due to restrictions from Soviet authorities. Jacques T**s expressed sadness over Margulis's absence during his address. He addresses Margulis' contributions to combinatorics, differential geometry, ergodic theory, dynamical systems, and discrete subgroups of Lie groups, emphasizing that the Fields Medal was mostly granted for his work in the latter area. He mentioned in [J. T**s, The work of Gregori Aleksandrovitch Margulis, Proceedings of the International Congress of Mathematicians, Helsinki, 1978 (Helsinki, 1980), 57-63.] that:-

"Already Poincaré wondered about the possibility of describing all discrete subgroups of finite covolume in a Lie group G. The profusion of such subgroups in
G=PSL_2(R) makes one at first doubt of any such possibility. However, PSL_2(R) was for a long time the only simple Lie group which was known to contain non-arithmetic discrete subgroups of finite covolume, and further examples discovered in 1965 by Makarov and Vinberg involved only few other Lie groups, thus adding credit to conjectures of Selberg and Pyatetski-Shapiro to the effect that "for most semisimple Lie groups" discrete subgroups of finite covolume are necessarily arithmetic. Margulis's most spectacular achievement has been the complete solution of that problem and, in particular, the proof of the conjecture in question."

Margulis was soon able to leave the Soviet bloc and, in 1979, he was able to spend three months at the University of Bonn. Between 1988 and 1991 Margulis made a number of visits to the Max Planck Institute in Bonn, to the Institut des Hautes Études and to the Collège de France, to Harvard and to the Institute for Advanced study in Princeton. From 1991 he has held a chair at Yale University.

The Oppenheim conjecture was made in 1929 and concerns values of indefinite irrational quadratic forms at integer points. Early work was based on results of Jarnik and Walfisz. In the 1940s Davenport and Heilbronn contributed by proving special cases and in 1946 Watson extended their results showing the conjecture to be true for further special cases. Margulis proved the full conjecture in 1986 and gives a beautiful survey of the work leading to this solution in [G .A. Margulis, Oppenheim conjecture, in M Atiyah and D Iagolnitzer (eds.), Fields Medallists Lectures (Singapore, 1997), 272-327.].

Margulis has garnered numerous accolades for his contributions, including the Fields Medal, the Medal of the Collège de France (1991), and honorary membership in the American Academy of Arts and Science in the same year. He received the Humboldt Prize in 1995 and became a member of the Tata Institute of Fundamental Research in 1996. Additionally, he was awarded the Lobachevsky International Prize by the Russian Academy of Sciences and has been elected to the United States National Academy of Sciences. In 2005 he was awarded the Wolf Prize for Mathematics:-

"... for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics and measure theory."

(Source: MacTutor)

Enrico Bombieri (born 26 November 1940) - The Fields Medalist of 1974

Enrico Bombieri was awarded the Fields Medal at the International Congress of Mathematicians in Vancouver in 1974 in recognition of his outstanding contributions. The award honored his major work on prime number theory, univalent functions and the local Bieberbach conjecture, functions of several complex variables, as well as partial differential equations and minimal surfaces, particularly his work on Sergei Bernstein’s problem in higher dimensions.

K. Chandrasekharan highlights Bombieri’s significant contributions to the theory of prime distribution, univalent functions and the local Bieberbach conjecture, and functions of several complex variables. He writes

"First among Bombieri's achievements is his remarkable theorem on the distribution of primes in arithmetical progressions, which is obtained by an application of the methods of the large sieve."

The large sieve method was first introduced by Linnik in 1941 while addressing problems posed by Vinogradov. Given an arithmetic progression, the large sieve gives information about the distribution of an arbitrary finite set of integers. Rényi further developed Linnik’s ideas in 1950, and in 1965, Klaus Roth and Enrico Bombieri independently refined Rényi’s results. Bombieri later used his improved large sieve technique to establish what is now known as Bombieri’s mean value theorem, which deals with the distribution of prime numbers in arithmetic progressions.

In 1966, Bombieri was appointed to a professorship at the University of Pisa. During this period, he became interested in the work of De Giorgi and his school of geometric measure theory at the Scuola Normale Superiore in Pisa. Their research focused on Plateau-type problems in spaces of dimension higher than three.

Bombieri was awarded the Balzan International Prize in 1980 and was elected a foreign member of the French Academy of Sciences in 1984. He now works in the United States. In 1996, he was elected to the National Academy of Sciences, and the citation for his election stated:

"Bombieri is one of the world's most versatile and distinguished mathematicians. He has significantly influenced number theory, algebraic geometry, partial differential equations, several complex variables, and the theory of finite groups. His remarkable technical strength is complemented by an unerring instinct for the crucial problems in key areas of mathematics."

In addition to the honors mentioned earlier, Bombieri received the Feltrinelli Prize in 1976, the Cavaliere di Gran Croce al Merito della Repubblica Italiana in 2002, and the Premio Internazionale Pitagora from the City of Crotone in 2006. In January 2008, he was jointly awarded the Doob Prize with Walter Gubler at the 114th Annual Meeting of the American Mathematical Society in San Diego, in recognition of their book Heights in Diophantine Geometry, which they co-authored. The citation for the prize reads as follows:

"The book is a research monograph on all aspects of Diophantine geometry, both from the perspective of arithmetic geometry and of transcendental number theory. ... One gets the sense that every lemma, every theorem, every remark has been carefully considered, and every proof has been thought through in every detail. There are well-chosen illuminating examples throughout every chapter. The book is a masterpiece in terms of its original approach, its unrivalled comprehensiveness, and the sheer elegance of the exposition. There can be no doubt that this book will become the basis for the future development of this central subject of modern mathematics."

(Source: MacTutor)
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