Math Labs

Grigory Margulis

Native spelling: Григорий Маргулис

fl. 1946, Moscow, Soviet Union

AlgebraDifferential equations and dynamical systemsCombinatorics and discrete mathematicsFields Medal 1978Wolf Prize in Mathematics 2005Abel Prize 2020

Russian-American mathematician who founded the theory of lattices in semisimple Lie groups (arithmeticity, superrigidity) and pioneered the use of ergodic theory and expander graphs across number theory, dynamics and combinatorics.

Grigory Aleksandrovich Margulis was born on 27 February 1946 in Moscow. A prodigy at mathematics olympiads, he studied at Moscow State University, where Yakov Sinai supervised his doctorate (1970) on ergodic properties of group actions, and where the schools of Kolmogorov, Gelfand and Sinai formed his mathematical outlook. He spent his Soviet career at the Institute for Problems of Information Transmission, since the establishment universities remained largely closed to him.

In the 1970s Margulis transformed the theory of discrete subgroups of Lie groups: he proved superrigidity and arithmeticity for lattices in semisimple groups of higher real rank, showing they always come from integer points of algebraic groups. In the 1980s he produced the first explicit constructions of expander graphs and of Ramanujan graphs, objects now central to computer science. His 1987 resolution of the Oppenheim conjecture — proving that a non-degenerate indefinite quadratic form in at least three variables takes values dense in the reals at integer points — opened the field now called homogeneous dynamics.

He emigrated to the United States in 1991, joining Yale University, where he is Erastus L. DeForest Professor of Mathematics, Emeritus. His influence was recognized by the Fields Medal in 1978, the Wolf Prize in 2005 (shared with Sergei Novikov) and the Abel Prize in 2020, awarded jointly with Hillel Furstenberg for pioneering the use of probability and dynamics in group theory, number theory and combinatorics.

Workplaces: Moscow State University, Institute for Problems of Information Transmission, Moscow, Yale University

Russia, United States

Award-winning work

Fields Medal 1978For his contributions to the theory of discrete subgroups of Lie groups. Margulis proved that a lattice in a semisimple Lie group of real rank at least two is arithmetic — solving Selberg's and Piatetski-Shapiro's arithmeticity conjecture — and established superrigidity: every finite-dimensional representation of such a lattice extends to a representation of the ambient Lie group. Together these results imply that all irreducible lattices in rank at least two arise from integer points of algebraic groups and that their actions are rigid in a deep sense, an outcome that also proved, among other consequences, the Oppenheim conjecture on values of indefinite quadratic forms. Although named a laureate in 1978, he was not permitted by Soviet authorities to attend the congress in Helsinki.

Contributions, linked to the library

Library topicErgodic theoryPioneered homogeneous dynamics: his proof of the Oppenheim conjecture showed that unipotent flows can solve Diophantine problems.Library topicLie groups and Lie algebrasArithmeticity and superrigidity theorems describe all lattices in semisimple Lie groups of rank at least two.Great problemLittlewood conjectureHis work on unipotent flows led to the measure-rigidity methods used by Einsiedler–Katok–Lindenstrauss for partial progress on Littlewood's conjecture.