Elon Lindenstrauss

Elon Lindenstrauss
Lindenstrauss in Berkeley, 2014
Elon Lindenstrauss
Born	1 8, 1970
Birthplace	Jerusalem, Israel
Nationality	Israeli
Occupation	Mathematician
Employer	Hebrew University of Jerusalem, Institute for Advanced Study
Known for	Ergodic theory, number theory, homogeneous dynamics, Fields Medal (2010)
Education	PhD, Hebrew University of Jerusalem
Awards	Fields Medal (2010), Fermat Prize (2009), Leonard Eisenbud Prize (2010)
Website	[http://www.math.huji.ac.il/~elon/ Official site]

Elon Lindenstrauss (Template:Lang-he; born August 1, 1970) is an Israeli mathematician who works at the intersection of ergodic theory, number theory, and dynamical systems. Born and raised in Jerusalem into a family with deep roots in Israeli mathematics, Lindenstrauss became the first Israeli to receive the Fields Medal, which he was awarded in 2010 at the International Congress of Mathematicians in Hyderabad, India, for his results on measure rigidity in ergodic theory and their applications to number theory. His work has demonstrated how the study of dynamical systems — mathematical models describing how points in a space evolve over time — can yield profound insights into problems in number theory that had resisted other approaches for decades. Lindenstrauss has held professorships at Princeton University and the Hebrew University of Jerusalem, and in 2024 was appointed a permanent faculty member in the School of Mathematics at the Institute for Advanced Study. His contributions have shaped the modern field of homogeneous dynamics and opened new pathways for applying ergodic-theoretic methods to long-standing arithmetic questions.

Early Life

Elon Lindenstrauss was born on August 1, 1970, in Jerusalem, Israel. He grew up in a family with significant connections to Israeli academic life and mathematics. His father, Joram Lindenstrauss, was a distinguished mathematician at the Hebrew University of Jerusalem who made foundational contributions to functional analysis and Banach space theory. His mother, Naomi Lindenstrauss, is also a mathematician. Growing up in this intellectually stimulating environment, Lindenstrauss was exposed to mathematical thought from an early age.

As a young student, Lindenstrauss showed exceptional aptitude in mathematics. He participated in the International Mathematical Olympiad (IMO), which is one of the most competitive mathematics competitions for pre-university students worldwide. His performance at the IMO marked him as a young mathematician of considerable promise.

His family connections to mathematics extended beyond his parents. His brother-in-law is Michael J. Larsen, a mathematician, and his nephew Daniel Larsen has also become known in the field of mathematics. This extended mathematical lineage underscores the rich intellectual tradition from which Lindenstrauss emerged.

Like many Israeli citizens, Lindenstrauss served in the Israel Defense Forces. He served in an intelligence unit, an experience common among young Israelis before entering university. Following his military service, he pursued his academic career at the Hebrew University of Jerusalem, the same institution where his father had built his career.

Education

Lindenstrauss completed all three of his academic degrees — Bachelor of Science, Master of Science, and Doctor of Philosophy — at the Hebrew University of Jerusalem. He pursued his doctoral studies under the supervision of Benjamin Weiss, a prominent mathematician known for his work in ergodic theory, probability, and dynamical systems. Weiss's influence was formative in guiding Lindenstrauss toward the problems in ergodic theory and dynamical systems that would define his career.

Lindenstrauss completed his PhD in 1999. His doctoral work laid the groundwork for his subsequent research into measure rigidity and its applications, establishing him as a promising young researcher in the field of ergodic theory. The Hebrew University, with its strong tradition in mathematics and its Einstein Institute of Mathematics, provided a fertile environment for his intellectual development.

Career

Early Academic Career

Following the completion of his doctorate, Lindenstrauss held several postdoctoral and visiting positions that allowed him to develop and expand his research program. He spent time at leading mathematical institutions, building collaborations that would prove central to his most important work. His early research focused on extending the understanding of measure rigidity — the phenomenon whereby certain dynamical systems have far fewer invariant measures than might be naively expected — and applying these ideas to problems in number theory and mathematical physics.

Princeton University

In 2004, Lindenstrauss joined the faculty of Princeton University as a professor of mathematics. Princeton provided a world-class environment for mathematical research, and Lindenstrauss became a central figure in the university's mathematics department. At Princeton, he continued his work on the rigidity properties of diagonalizable actions on locally homogeneous spaces, a subject that sits at the crossroads of several major areas of mathematics.

During his time at Princeton, Lindenstrauss produced some of his most celebrated work. He developed new methods and proved results that had significant implications for number theory, including advances related to the Littlewood conjecture and the quantum unique ergodicity conjecture. His approach was distinctive in that it brought the tools of ergodic theory — a branch of mathematics originally developed to study statistical properties of deterministic systems — to bear on problems that had traditionally been approached using algebraic or analytic number-theoretic methods.

Peter Sarnak, a colleague at Princeton, described Lindenstrauss's work as having "striking applications to number theory." According to Sarnak, quoted by Princeton University, "Elon has introduced many novel techniques to the field, and his insights have been deep and penetrating."

Hebrew University of Jerusalem

In 2009, while maintaining his position at Princeton, Lindenstrauss was appointed a professor at the Einstein Institute of Mathematics at the Hebrew University of Jerusalem. This appointment represented a return to his academic origins and to the institution where he had earned all of his degrees. The Hebrew University had long been a center of mathematical excellence in Israel, and Lindenstrauss's presence further strengthened its standing in the areas of ergodic theory and dynamical systems.

At the Hebrew University, Lindenstrauss continued to develop his research program in homogeneous dynamics. The field of homogeneous dynamics is concerned with the behavior of flows on spaces that arise naturally in number theory, such as the space of lattices in Euclidean space. As Lindenstrauss described in work highlighted by the European Research Council, the approach involves studying arithmetic objects like integer points using the theory of dynamical systems — a technique that has yielded results that are difficult or impossible to obtain by purely number-theoretic means.

Institute for Advanced Study

In 2024, Lindenstrauss was appointed a permanent faculty member in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey. The Institute for Advanced Study, founded in 1930, is one of the world's foremost centers for theoretical research and has been home to some of the most significant mathematicians and scientists in history, including Albert Einstein, John von Neumann, and Kurt Gödel. A permanent appointment at the Institute is among the highest distinctions in academic mathematics and reflects the enduring significance of Lindenstrauss's contributions to the field.

Research Contributions

Lindenstrauss's mathematical work centers on the rigidity of measures invariant under group actions on homogeneous spaces and the application of these results to problems in number theory. Several strands of his research have been particularly influential.

Measure Rigidity and Littlewood Conjecture

One of Lindenstrauss's major contributions involves work related to the Littlewood conjecture, a long-standing problem in Diophantine approximation posed in the 1930s. The conjecture concerns the simultaneous approximation of two real numbers by rationals and states, roughly, that for any two real numbers, the product of the errors in their rational approximations can be made arbitrarily small. Working with Manfred Einsiedler and Anatole Katok, Lindenstrauss proved that the set of possible exceptions to the Littlewood conjecture has Hausdorff dimension zero — meaning that even if exceptions exist, they form an extraordinarily small and sparse set. This result was achieved through an innovative application of measure classification techniques to the action of diagonal matrices on the space of lattices, a quintessential problem in homogeneous dynamics.

Arithmetic Quantum Unique Ergodicity

Another landmark result by Lindenstrauss concerns the quantum unique ergodicity (QUE) conjecture, proposed by Zeev Rudnick and Peter Sarnak in 1994. The conjecture addresses the distribution of quantum states on negatively curved surfaces — specifically, whether eigenfunctions of the Laplacian on such surfaces become equidistributed in the high-energy limit. Lindenstrauss proved a version of this conjecture for arithmetic hyperbolic surfaces, establishing what is known as arithmetic quantum unique ergodicity. This result sits at the interface of ergodic theory, number theory, and mathematical physics, and it represents one of the most significant applications of dynamical systems methods to problems arising from quantum mechanics.

The Fields Medal citation recognized Lindenstrauss specifically "for his results on measure rigidity in ergodic theory, and their applications to number theory."

Homogeneous Dynamics

More broadly, Lindenstrauss has been a leading figure in the development of homogeneous dynamics as a unified field. Homogeneous dynamics studies the action of subgroups of Lie groups on quotient spaces, and it has connections to a wide range of mathematical disciplines including number theory, geometry, and combinatorics. Lindenstrauss's work has shown how classification theorems for invariant measures — results that determine which measures are preserved by a dynamical system — can serve as powerful tools for answering questions that originate far from the traditional domain of dynamical systems.

As described by the European Research Council, Lindenstrauss's approach involves using the theory of dynamical systems to study arithmetic objects such as integer points, yielding results that illuminate fundamental questions in number theory.

Personal Life

Lindenstrauss was born and raised in Jerusalem into a mathematical family. His father, Joram Lindenstrauss (1936–2012), was a prominent Israeli mathematician known for his work in functional analysis and geometric aspects of Banach space theory at the Hebrew University of Jerusalem. His mother, Naomi Lindenstrauss, is also a mathematician. This family background in mathematics provided a strong intellectual foundation for Lindenstrauss's own career.

Lindenstrauss's brother-in-law is the mathematician Michael J. Larsen, and his nephew is Daniel Larsen, who has attracted attention for his own mathematical work at a young age.

Lindenstrauss has maintained professional ties to both Israel and the United States throughout his career, holding positions simultaneously in both countries. He has been based at institutions in Princeton, New Jersey, and Jerusalem, reflecting his connections to both the Israeli and American mathematical communities.

Recognition

Fields Medal (2010)

Lindenstrauss received the Fields Medal in 2010 at the International Congress of Mathematicians held in Hyderabad, India. The Fields Medal, awarded every four years by the International Mathematical Union to mathematicians under the age of 40, is often described as the highest honor in mathematics. Lindenstrauss was cited "for his results on measure rigidity in ergodic theory, and their applications to number theory." He was the first Israeli mathematician to receive the Fields Medal, an achievement that was noted with considerable attention in Israel.

The award was covered internationally, and Israeli media reported extensively on the achievement. The Jerusalem Post announced the award under the headline "Israeli wins 'Nobel' of Mathematics," referencing the common comparison between the Fields Medal and the Nobel Prize. Lindenstrauss was 40 years old at the time of the award, at the upper boundary of the age eligibility requirement.

Other Fields Medal recipients in 2010 included Ngô Bảo Châu, Stanislav Smirnov, and Cédric Villani.

Fermat Prize (2009)

In 2009, Lindenstrauss received the Fermat Prize for Mathematics Research for his work in number theory. The Fermat Prize is awarded biennially by the Institut de Mathématiques de Toulouse for contributions to fields where the research interests of Pierre de Fermat were influential. Princeton University announced the award, noting that the prize recognized Lindenstrauss's contributions to number theory through his work in ergodic theory and dynamical systems.

Leonard Eisenbud Prize (2010)

Lindenstrauss received the Leonard Eisenbud Prize for Mathematics and Physics from the American Mathematical Society in 2010, jointly with Manfred Einsiedler and Anatole Katok, for their work on the Littlewood conjecture.

Other Recognition

Lindenstrauss has been affiliated with the Simons Institute for the Theory of Computing at the University of California, Berkeley. He has also received funding from the European Research Council for his work in homogeneous dynamics.

Legacy

Lindenstrauss's mathematical contributions have had a transformative effect on the field of homogeneous dynamics and on the application of ergodic-theoretic methods to number theory. His work has demonstrated that measure rigidity results — theorems that classify the invariant measures for certain group actions — can serve as tools of remarkable power for solving problems in arithmetic and mathematical physics. This approach has opened new research directions and inspired a generation of mathematicians working at the interface of dynamics and number theory.

His proof of arithmetic quantum unique ergodicity resolved a significant case of a conjecture that had been a central open problem in mathematical physics and spectral geometry. The result established a definitive connection between the algebraic structure of arithmetic surfaces and the equidistribution properties of quantum states, linking deep ideas from number theory, representation theory, and quantum mechanics.

The work on the Littlewood conjecture, carried out with Einsiedler and Katok, showed that measure classification techniques could yield results in Diophantine approximation that were beyond the reach of classical methods. While the Littlewood conjecture itself remains unresolved in full generality, the proof that the set of exceptions has Hausdorff dimension zero stands as one of the strongest results achieved in this area.

As the first Israeli recipient of the Fields Medal, Lindenstrauss's achievement also had symbolic importance for the Israeli mathematical community. Tablet Magazine noted that his Fields Medal "marks Jewish achievement in the field" of mathematics and reflected the strength of Israeli contributions to the mathematical sciences. His career trajectory — from student at the Hebrew University, through positions at Princeton, and returning to the Hebrew University as a professor — illustrates the close connections between the Israeli and international mathematical communities.

Lindenstrauss's appointment as a permanent faculty member at the Institute for Advanced Study in 2024 further solidified his standing as one of the leading mathematicians of his generation.