**Gregori Margulis** was educated at Moscow High School, graduating in 1962. In that year he began his undergraduate studies at Moscow University and he was awarded his first degree in 1967. Margulis remained at Moscow University for his postgraduate studies.

He showed great potential as a mathematician and the first important award which he won was during his time as a postgraduate student when he received the young mathematicians prize from the Moscow Mathematical Society in 1968. Margulis completed his graduate studies in 1970 and he was awarded the degree of Candidate of Science for a thesis *On some problems in the theory of U-systems*.

After being awarded the Candidate of Science degree (the equivalent of a British or American Ph.D.), Margulis began to work in the Institute for Problems in Information Transmission. He was a Junior scientific worker there from 1970 to 1974 when he was promoted to Senior scientific worker. He held this post until 1986 when he was promoted again, this time to Leading scientific worker.

International honour was given to Margulis in 1978 when he was awarded a Fields Medal at the International Congress at Helsinki. However it was not a happy occasion for Margulis who was not permitted by the Soviet authorities to travel to Helsinki to receive the Medal. Tits, delivering the address [7] spoke of his sadness that Margulis could not be present:-

... I cannot but express my deep disappointment - no doubt shared by many people here - in the absence of Margulis from this ceremony. In view of the symbolic meaning of this city of Helsinki, I had indeed grounds to hope that I would have a chance at last to meet a mathematician whom I know only through his work and for whom I have the greatest respect and admiration.

Perhaps Tits' comment about 'symbolic meaning' should be explained. He delivered the address in the Finlandia Hall in Helsinki where Margulis should have received the Medal and where the Helsinki Accords had been signed on 1 August 1975. This major agreement was signed at the end of the first Conference on Security and Cooperation in Europe. The Helsinki Accords, signed by all the countries of Europe (excluding Albania) and by the United States and Canada, were designed to reduce the cold war tension by accepting the European boundaries as they then were.

Tits talks in [7] about the range of Margulis's work in combinatorics, differential geometry, ergodic theory, dynamical systems and discrete subgroups of Lie groups. The award of the Fields Medal was mainly for his work on this latter topic:-

Already Poincaré wondered about the possibility of describing all discrete subgroups of finite covolume in a Lie group G. The profusion of such subgroups inG=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 in1965by 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 [3]. There Margulis explains that:-

The different approaches to this and related conjectures(and theorems)involve analytic number theory, the theory of Lie groups and algebraic groups, ergodic theory, representation theory, reduction theory, geometry of numbers and some other topics.

Margulis has received many honours for his work. In addition to the Fields Medal he has been awarded the Medal of the Collège de France (1991) and in the same year he was elected an honorary member of the American Academy of Arts and Science. In 1995 he received the Humboldt Prize and in 1996 he was honoured by election as a member of the Tata Institute of fundamental research.

Margulis has also been awarded the Lobachevsky International Prize of 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.

An article in the May 2005 part of the *Notices* of the American Mathematical Society explains the work which led to the award:-

At the centre of the work of Gregory Margulis lies his proof of the Selberg-Piatetskii-Shapiro Conjecture, affirming that lattices in higher rank Lie groups are arithmetic, a question whose origins date back to Poincaré. This was achieved by a remarkable tour de force, in which probabilistic ideas revolving around a noncommutative version of the ergodic theorem were combined with p-adic analysis and with algebraic geometric ideas showing that "rigidity" phenomena, earlier established by Margulis and others, could be formulated in such a way("super-rigidity")as to imply arithmeticity. This work displays stunning technical virtuosity and originality, with both algebraic and analytic methods. The work has subsequently reshaped the ergodic theory of general group actions on manifolds.

In a second tour de force, Margulis solved the1929Oppenheim Conjecture, stating that the set of values at integer points of an indefinite irrational nondegenerate quadratic form in more than three variables is dense inR.^{n}. This had been reduced (by Rhagunathan) to a conjecture about unipotent flows on homogeneous spaces, proved by Margulis. This method transformed to this ergodic setting a family of questions till then investigated only in analytic number theory

A third dramatic breakthrough came when Margulis showed that Kazhdan's "Property T"(known to hold for rigid lattices)could be used in a single arithmetic lattice construction to solve two apparently unrelated problems. One was the solution to a problem posed by Rusiewicz, about finitely additive measures on spheres and Euclidean spaces. The other was the first explicit construction of infinite families of expander graphs of bounded degree, a problem of practical application in the design of efficient communication networks. Margulis's work is characterized by extraordinary depth, technical power, creative synthesis of ideas and methods from different areas of mathematics, and a grand architectural unity of its final form. Though his work addresses deep unsolved problems, his solutions are housed in new conceptual and methodological frameworks of broad and enduring application. He is one of the mathematical giants of the last half century.

In 2008 the *Pure and Applied Mathematics Quarterly* produced a Special Issue in honour of Margulis. The Introduction states:-

Gregory Margulis is a mathematician of great depth and originality. Besides his celebrated results on super-rigidity and arithmeticity of irreducible lattices of higher rank semisimple Lie groups, and the solution of the Oppenheim conjecture on values of irrational indefinite quadratic forms at integral points, he has also initiated many other directions of research and solved a variety of famous open problems.

Finally we end this biography by quoting from Tits [7]:-

Margulis has completely or almost completely solved a number of important problems in the theory of discrete subgroups of Lie groups, problems whose roots lie deep in the past and whose relevance goes far beyond that theory itself. It is not exaggerated to say that, on several occasions, he has bewildered the experts by solving questions which appeared to be completely out of reach at the time. He managed that through his mastery of a great variety of techniques used with extraordinary resources of skill and ingenuity. The new and most powerful methods he has invented have already had other important applications besides those for which they were created and, considering their generality, I have no doubt that they will have many more in the future.

**Article by:** *J J O'Connor* and *E F Robertson*

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