Elon Lindenstrauss

Native spelling: אילון לינדנשטראוס

fl. 1970, Jerusalem, Israel

Differential equations and dynamical systemsArithmetic and number theoryAnalysisFields Medal 2010

Israeli mathematician whose measure-rigidity results in ergodic theory gave the strongest progress on Littlewood's conjecture and resolved quantum unique ergodicity on arithmetic surfaces; Fields Medal 2010.

Elon Lindenstrauss was born in Jerusalem in 1970, the son of the mathematician Joram Lindenstrauss. He studied mathematics and physics at the Hebrew University of Jerusalem, and received his PhD there in 1999 under Benjamin Weiss. After postdoctoral positions at Stanford University and the Institute for Advanced Study, and a period as a professor at Princeton University, he returned to the Hebrew University, where he is a professor at the Einstein Institute of Mathematics.

Lindenstrauss works at the meeting point of ergodic theory and Diophantine approximation. The measure rigidity program, initiated by Furstenberg, asserts that invariant measures for multi-dimensional dynamical systems should be algebraic; Lindenstrauss, largely with Manfred Einsiedler and Anatole Katok, proved decisive partial results. Their 2006 theorem implies that the exceptional set in Littlewood's conjecture — pairs of real numbers that cannot be simultaneously approximated well by rationals with a common denominator — has Hausdorff dimension zero. This remains the best evidence for the conjecture short of a full proof.

His further work ranges from the quantum unique ergodicity conjecture of Rudnick and Sarnak (proved for arithmetic surfaces with Soundararajan) to effective equidistribution on homogeneous spaces. He received the Fields Medal in 2010 in Hyderabad, alongside Ngô Bảo Châu, Stanislav Smirnov and Cédric Villani, and won the Nemmers Prize in Mathematics in 2016.

Workplaces: Hebrew University of Jerusalem, Princeton University

Israel, United States

Award-winning work

Fields Medal 2010The medal cites his results on measure rigidity in ergodic theory and their applications to number theory. In a landmark collaboration with Manfred Einsiedler and Anatole Katok, Lindenstrauss proved that under positive entropy, invariant measures for higher-rank torus actions are severely restricted, implying that the set of pairs of real numbers that could violate Littlewood's conjecture on simultaneous Diophantine approximation has Hausdorff dimension zero — by far the strongest partial result on that conjecture. With Kannan Soundararajan he also proved quantum unique ergodicity for arithmetic hyperbolic surfaces, settling the problem of how eigenfunctions of the Laplacian distribute themselves in the quantum limit. These works turned measure rigidity into a versatile tool linking dynamics, number theory and mathematical physics.

Contributions, linked to the library

Great problemLittlewood conjectureWith Einsiedler and Katok, proved that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero.Library topicErgodic theoryCentral figure of the measure-rigidity program for higher-rank dynamical systems.