### Gregory Margulis

Born 24 February 1946, Moscow, USSR (Russia)

Fields Medal (1978) “for his work on combinatorics, differential geometry, ergodic theory, the theory of dynamical systems and the theory of discrete subgroups of real and p-adic Lie groups. The last aspect covered his main results by the time of the award.”

Grigori Margulis was born in Moscow, where he received his schooling. His father was a mathematician mostly involved in mathematical education. His mother worked as a librarian. Margulis began his undergraduate education in 1962 at Moscow State University. His first research paper was published in 1966 while he was still an undergraduate student. He graduated in 1967, but remained at the University as a graduate student until 1970. During this time he won the Moscow Mathematical Society’s prize for young mathematicians (1968). In 1970 Margulis received a ‘candidate of sciences’ degree, which is equivalent to a Ph.D. His thesis was about problems in the theory of a class of dynamical systems which are now called Anosov systems. In the following years, Margulis took a job at the Moscow Institute for problems in information transmission of the Soviet Academy of Sciences, as a junior research fellow (1970-74), senior research fellow (1974-86), and as a leading research fellow (1986-91). When Margulis was awarded the Fields Medal in 1978, he was not permitted to attend the award ceremony due to the opposition of the top soviet mathematical establishment, despite the historical location: the International Congress of Mathematicians was held in Helsinki, recently made famous by the signature of the ‘Helsinki Accords’ after the international peace negotiations of the CSCE there. It wasn’t until 1979 that Margulis was allowed to come to the West for the first time. It was a three-month visit arranged by his colleague Friedrich Hirzebruch. During that visit Jacques Tits came to Bonn and at a small ceremony presented the Fields Medal award.

Between 1988 and 1991 Margulis visited the Max Planck Institute for Mathematics in Bonn, the Institut des Hautes Études and the Collège de France, Harvard University, and the Institute for Advanced Study at Princeton. In 1991, he joined the faculty of Yale University, where he currently is Erastus L. DeForest Professor of Mathematics.

He is married to Raisa Margulis, and he has one son Boris Margulis.

Grigori Margulis received the Medal of the Collège de France in 1990 in connection with the visiting professorship. In 1991 he was elected a Foreign Honorary Member the American Academy of Arts and Sciences. He was awarded the Humboldt Research Award (1995), Lobachevsky Prize (1997), the Wolf prize (2005), and the Dobrushin International Prize (2011). He was elected a Honorary Fellow of the Tata Institute of Fundamental Research (1996), a member of the National Academy of Sciences of the USA (2001), a Fields Institute Fellow (2011), and a Fellow of the American Mathematical Society (2012). He received honorary doctoral degrees from the University of Bielefeld (1999), Ecole Normale Superiere, Paris (ENS) (2010), the University of York (2011), and the University of Lyon (2013).

Over the decades Grigori Margulis has, with astonishing productivity, worked on an unusually broad spectrum of research topics. He has built bridges between number theory, ergodic theory, dynamical systems, combinatorics, algebraic geometry, and the theory of Lie groups and its discrete subgroups.

He became famous in the 1970s with an important result on the structure of Lie groups, which — in contrast to discrete groups — describe continuous symmetries. Examples of Lie groups are rotations in a plane or all maps in space that preserve length (ie objects are essentially only moved or rotated). Lie groups can also contain discrete subgroups: the planar rotations, for example, contain planar rotations by multiples of 90 degrees. Jules Henri Poincaré (1854-1912) had already speculated whether it is possible to classify all discrete subgroups (with certain additional properties) of a Lie group, and Atle Selberg (1917-2007) and Ilya Piatetski-Shapiro (1929-2009) later refined his conjecture. Margulis showed not only that such a classification is actually possible, but also what it looks like, and he developed from this a characterisation of the ‘arithmetic groups’. For his proof, Margulis used a massive mathematical apparatus, combining for example means from ergodic theory with p-adic analysis and ideas from algebraic geometry.

Margulis’s second major proof was in 1986 and concerned a conjecture of Victor Alexander Oppenheim (1903-1997) from 1929. Margulis showed that Oppenheim was right in his assumption that irrational indefinite quadratic forms in three or more variables can take arbitrarily small values at nonzero integral points. An example of such a form is $x^2 + y^2 – \sqrt{2} z^2$.

Margulis also achieved a number of important results in other areas including measure theory and combinatorics. For example, as an application of the theory of discrete subgroups of real and p-adic Lie groups he created a method to construct ‘expander graphs’, which are today not only important in communication technology, but are also an exciting field of research in themselves.