Gregori Margulis

Gregori Margulis
Born	Grigory Aleksandrovich Margulis 24 2, 1946
Birthplace	Moscow, Soviet Union
Nationality	American, Russian
Occupation	Mathematician
Employer	Yale University
Known for	Superrigidity theorem, Arithmeticity theorem, work on lattices in Lie groups, ergodic theory
Education	Moscow State University (PhD)
Awards	Fields Medal (1978), Wolf Prize in Mathematics (2005), Abel Prize (2020)

Gregori Aleksandrovich Margulis (born February 24, 1946) is a Russian-American mathematician whose profound contributions to the theory of lattices in semisimple Lie groups, ergodic theory, and number theory have reshaped significant areas of modern mathematics. Born in Moscow during the Soviet era, Margulis emerged as one of the foremost mathematicians of his generation, earning the Fields Medal in 1978 at the age of 32 — though political restrictions imposed by the Soviet government prevented him from traveling to Helsinki to accept the award in person. Over the course of a career spanning more than five decades, his work has established deep and unexpected connections between dynamics, geometry, and number theory, yielding results that continue to influence research across multiple mathematical disciplines. In 2020, Margulis was awarded the Abel Prize, alongside Hillel Furstenberg, for their pioneering use of methods from probability and dynamics in group theory, number theory, and combinatorics. His other major honors include the Wolf Prize in Mathematics (2005), making him one of a select group of mathematicians to have received all three of the discipline's most prestigious awards: the Fields Medal, the Wolf Prize, and the Abel Prize.

Early Life

Grigory Aleksandrovich Margulis was born on February 24, 1946, in Moscow, in what was then the Soviet Union. He grew up in the Soviet capital during the postwar period, a time when the Soviet mathematical tradition was flourishing under the influence of prominent schools led by figures such as Andrey Kolmogorov and Israel Gelfand. From an early age, Margulis demonstrated exceptional mathematical aptitude. The Soviet system of mathematical education, which placed strong emphasis on olympiads and specialized secondary schools, provided a fertile environment for his talents.

Margulis entered Moscow State University, one of the premier centers for mathematical research in the Soviet Union, where he pursued his undergraduate and graduate studies. He came under the mentorship of Yakov Sinai and David Kazhdan, both of whom were leading figures in ergodic theory and representation theory, respectively. This intellectual environment proved decisive in shaping Margulis's research interests, orienting him toward the interplay between dynamical systems, group theory, and geometry that would define his career.

Despite his extraordinary mathematical talent, Margulis faced significant obstacles in the Soviet Union due to his Jewish heritage. The systemic antisemitism prevalent in Soviet academic institutions limited his career prospects within the country. He was unable to secure a position at Moscow State University or at the most prestigious Soviet research institutes, and instead worked at the Institute for Problems in Information Transmission, a research institution of the Soviet Academy of Sciences. These professional limitations, while constraining, did not diminish his mathematical output, and he continued to produce groundbreaking work throughout the 1970s and 1980s while based at this institute.

Education

Margulis completed his entire formal education at Moscow State University, one of the leading mathematical centers in the world during the Soviet era. He earned his undergraduate degree and subsequently his doctoral degree (the Soviet equivalent of a PhD) from the university's Department of Mechanics and Mathematics. His doctoral research, supervised by Yakov Sinai, focused on problems in ergodic theory and dynamical systems, areas that would remain central to his mathematical work. The rigorous training he received at Moscow State University, combined with the influence of the vibrant Soviet mathematical community, equipped Margulis with a broad toolkit spanning analysis, algebra, geometry, and probability — a breadth that would prove essential to his later interdisciplinary achievements.

Career

Early Work and the Fields Medal

Margulis's early research in the 1970s produced a series of results that transformed the understanding of discrete subgroups of Lie groups. His most celebrated early achievements were the superrigidity theorem and the arithmeticity theorem for lattices in semisimple Lie groups of rank greater than one. The superrigidity theorem established that homomorphisms from higher-rank lattices to algebraic groups are essentially determined by algebraic data, a result that had far-reaching consequences. The arithmeticity theorem, which followed as a corollary, proved that all irreducible lattices in semisimple Lie groups of real rank at least two are arithmetic — meaning they arise from number-theoretic constructions. This resolved a major conjecture that had been open for years and represented a landmark achievement in the theory of discrete groups.

These results, along with his other contributions, led to the award of the Fields Medal in 1978 at the International Congress of Mathematicians held in Helsinki, Finland. The Fields Medal, often described as the most prestigious award in mathematics, is given every four years to mathematicians under the age of 40 for outstanding achievements. However, Margulis was unable to attend the ceremony to receive the prize in person, as the Soviet authorities denied him permission to travel abroad. This episode highlighted the political constraints under which many Soviet scientists and mathematicians operated during the Cold War. The medal was presented to him later through other arrangements.

Contributions to Ergodic Theory and Number Theory

Beyond his work on lattices and Lie groups, Margulis made fundamental contributions to ergodic theory and its applications to number theory. One of his most celebrated results in this area was his proof of the Oppenheim conjecture in 1987. This conjecture, posed by Alexander Oppenheim in 1929, concerned the values taken by indefinite quadratic forms at integer points. Specifically, it asserted that for an indefinite irrational quadratic form in three or more variables, the values at integer points are dense in the real numbers. Margulis proved this conjecture using methods from the theory of unipotent flows on homogeneous spaces — an approach that was unexpected and highly original, as the problem had traditionally been considered one of pure number theory.

The proof of the Oppenheim conjecture represented a paradigm shift in the application of dynamical methods to number-theoretic problems. It demonstrated that techniques from ergodic theory and the study of group actions could resolve longstanding questions in the theory of Diophantine approximation. This work catalyzed an entire field of research at the intersection of dynamics and number theory, and many subsequent results by other mathematicians have built upon the methods Margulis introduced.

Margulis also contributed significantly to the study of unipotent flows on homogeneous spaces more broadly. His work in this area, along with later developments by Marina Ratner and others, led to a comprehensive understanding of the behavior of orbits under unipotent group actions — results that have had applications ranging from number theory to mathematical physics.

Expander Graphs and Combinatorics

In an influential contribution outside his primary areas of research, Margulis provided one of the first explicit constructions of expander graphs in the 1970s. Expander graphs are sparse graphs with strong connectivity properties, and they have become fundamental objects in theoretical computer science, coding theory, and combinatorics. Prior to Margulis's work, the existence of expander graphs had been established through probabilistic (non-constructive) arguments. Margulis's construction, which used the theory of group representations and the properties of lattices in Lie groups, was among the first to provide an explicit family of such graphs. This contribution demonstrated the power of deep algebraic and analytic methods in addressing problems in discrete mathematics and computer science.

Move to the United States and Career at Yale

Following the dissolution of the Soviet Union, Margulis was able to pursue opportunities outside Russia. He eventually moved to the United States, where he joined the faculty of Yale University in New Haven, Connecticut. At Yale, he became the Erastus L. De Forest Professor of Mathematics, a distinguished chair reflecting his stature in the mathematical community. His position at Yale provided him with a stable and supportive environment in which to continue his research, and he became a central figure in the university's mathematics department.

At Yale, Margulis continued to produce important research while also mentoring a new generation of mathematicians. His presence contributed to Yale's strength in areas including number theory, ergodic theory, and geometric group theory. One notable indication of the department's strength in related fields came in 2013, when Hee Oh, a mathematician working in areas closely connected to Margulis's research on dynamics and number theory, became the first Korean female mathematician to receive tenure at Yale. Oh's work on dynamics on homogeneous spaces and counting problems was influenced by the mathematical tradition that Margulis had helped establish.

The Abel Prize

In March 2020, the Norwegian Academy of Science and Letters announced that Margulis and Hillel Furstenberg would share the Abel Prize, one of the highest honors in mathematics, for "pioneering the use of methods from probability and dynamics in group theory, number theory, and combinatorics." The Abel Prize, established in 2003, is awarded annually and carries a monetary award of approximately 7.5 million Norwegian kroner. It has been described alongside the Fields Medal and the Wolf Prize as one of the three most prestigious awards in mathematics.

The citation for the 2020 Abel Prize recognized both laureates for having independently and in complementary ways demonstrated the power of probabilistic and dynamical methods in solving problems across a broad range of mathematical disciplines. For Margulis specifically, the prize acknowledged his deep contributions to the understanding of lattices in Lie groups, his proof of the Oppenheim conjecture, his construction of expander graphs, and his many other contributions that brought together ideas from different mathematical fields in novel and powerful ways.

The Abel Prize had previously been awarded to several other distinguished mathematicians, including Yves Meyer in 2017, Dennis Sullivan in 2022, Luis Caffarelli in 2023, and Michel Talagrand in 2024.

Personal Life

Margulis is a private individual who has generally kept his personal life out of public view. His Jewish heritage played a significant role in his early career, as the antisemitic policies of the Soviet academic establishment restricted his professional advancement despite his extraordinary achievements. The denial of his travel permission to attend the 1978 Fields Medal ceremony in Helsinki was one of the most publicly visible consequences of these restrictions.

After emigrating from the Soviet Union, Margulis settled in the United States, where he has been based at Yale University in New Haven, Connecticut. He became an American citizen while maintaining connections to the broader international mathematical community. Throughout his life, Margulis has been known for his modesty and his deep dedication to mathematical research, preferring to let his work speak for itself rather than seeking public attention.

Recognition

Margulis is one of a small number of mathematicians to have received all three of the most prestigious awards in the field: the Fields Medal, the Wolf Prize, and the Abel Prize.

Fields Medal (1978): Awarded at the International Congress of Mathematicians in Helsinki for his work on lattices in semisimple Lie groups, including the superrigidity and arithmeticity theorems. Margulis was 32 years old at the time of the award but was unable to attend the ceremony due to Soviet travel restrictions.

Wolf Prize in Mathematics (2005): The Wolf Prize, awarded by the Wolf Foundation in Israel, recognized Margulis for his contributions to algebra, particularly to the theory of lattices in semisimple Lie groups and applications to ergodic theory, representation theory, number theory, combinatorics, and measure theory.

Abel Prize (2020): Shared with Hillel Furstenberg, the Abel Prize recognized both mathematicians for "pioneering the use of methods from probability and dynamics in group theory, number theory, and combinatorics."

In addition to these three major prizes, Margulis has received numerous other honors throughout his career. He is a member of the National Academy of Sciences of the United States and has been elected to several other national and international academies. He has been invited to deliver major lectures at mathematical conferences worldwide.

Legacy

Gregori Margulis's mathematical legacy is substantial and multifaceted. His superrigidity and arithmeticity theorems for lattices in higher-rank semisimple Lie groups remain cornerstones of modern mathematics, having fundamentally altered the understanding of discrete subgroups and their relationship to arithmetic constructions. These results opened new avenues of research in geometric group theory, algebraic groups, and differential geometry, and they continue to inform contemporary mathematical work.

His proof of the Oppenheim conjecture using dynamical methods is often cited as a turning point in the application of ergodic theory to number theory. By showing that questions about values of quadratic forms at integer points could be answered through the study of orbit closures on homogeneous spaces, Margulis established a methodology that has been extended and refined by subsequent generations of mathematicians. The broader program of using dynamics on homogeneous spaces to address number-theoretic problems — sometimes referred to as the "Margulis method" — has become a central theme in modern analytic number theory and Diophantine approximation.

Margulis's construction of expander graphs has had a lasting impact outside pure mathematics, particularly in theoretical computer science and network design. The explicit construction of these graphs using representation-theoretic methods demonstrated that abstract mathematical tools could have concrete applications in applied and computational settings.

The award of the Abel Prize in 2020, shared with Hillel Furstenberg, underscored the significance of the broader research program that both mathematicians helped initiate — one in which probabilistic and dynamical perspectives are brought to bear on problems that span algebra, number theory, and combinatorics. The methods and results that Margulis developed over more than five decades of work continue to generate new research directions and to inspire mathematicians working at the interfaces of dynamics, geometry, algebra, and number theory.