Peter Sarnak

Peter Sarnak
Born	Peter Clive Sarnak 12/18/1953
Birthplace	Johannesburg, South Africa
Nationality	South African, American
Occupation	Mathematician
Title	Eugene Higgins Professor of Mathematics (Princeton University); Gopal Prasad Professor of Mathematics (Institute for Advanced Study)
Employer	Institute for Advanced Study, Princeton University
Known for	Analytic number theory, spectral theory, arithmetic geometry
Education	PhD, Stanford University
Awards	Wolf Prize in Mathematics (2024), Shaw Prize in Mathematical Sciences (2024), Sylvester Medal (2019)
Website	https://www.math.ias.edu/people/faculty/sarnak

Peter Clive Sarnak (born 18 December 1953) is a South African-born American mathematician whose work spans analytic number theory, spectral theory, automorphic forms, and arithmetic geometry. Born in Johannesburg, South Africa, Sarnak studied at the University of the Witwatersrand before completing his doctoral work at Stanford University under the supervision of Paul Cohen. He has held permanent faculty positions at some of the most prominent mathematical institutions in the world, including the Courant Institute of Mathematical Sciences at New York University, Stanford University, and Princeton University, where he has served as the Eugene Higgins Professor of Mathematics since 2002. Since 2007, he has also been a member of the permanent faculty at the School of Mathematics at the Institute for Advanced Study. A Fellow of the Royal Society and a recipient of numerous international prizes, Sarnak's contributions have shaped the modern understanding of connections between number theory, geometry, and mathematical physics. He has served as an editor of the Annals of Mathematics and has been involved in the adjudication of the Shaw Prize in Mathematical Sciences.

Early Life

Peter Clive Sarnak was born on 18 December 1953 in Johannesburg, South Africa. He grew up in Johannesburg during the era of apartheid, attending school in the city before pursuing his undergraduate education at the University of the Witwatersrand (commonly known as Wits), also located in Johannesburg. Sarnak completed his Bachelor of Science degree at Wits in 1974, followed by a Bachelor of Science Honours degree in 1975.

The mathematical environment at Wits during the 1970s provided Sarnak with a foundation in pure mathematics, and his early academic performance signaled exceptional promise. After completing his honours degree, Sarnak left South Africa to pursue graduate studies in the United States, a trajectory that would ultimately lead him to become one of the most influential mathematicians of his generation.

Education

Sarnak received his undergraduate training at the University of the Witwatersrand in Johannesburg, earning a BSc in 1974 and a BSc Honours in 1975. He then moved to the United States to pursue doctoral studies at Stanford University, where he worked under the supervision of Paul Cohen, the mathematician who had famously proved the independence of the continuum hypothesis from the standard axioms of set theory.

Sarnak completed his PhD at Stanford in 1980 with a dissertation titled Prime Geodesic Theorems. The thesis explored the distribution of prime geodesics on Riemann surfaces, a topic that elegantly connected ideas from number theory, differential geometry, and spectral theory. This work under Cohen's mentorship established the interdisciplinary approach that would characterize Sarnak's career, bridging areas of mathematics that had previously been studied in relative isolation.

Career

Early Academic Positions

Following the completion of his doctorate in 1980, Sarnak began his academic career at the Courant Institute of Mathematical Sciences at New York University, one of the leading centers for applied and pure mathematics in the United States. His time at the Courant Institute allowed him to develop further the themes of his doctoral research while expanding his network of collaborators in the mathematical community.

Sarnak subsequently moved to Stanford University, where he held a faculty position and continued his research into the interplay between number theory and spectral geometry. At Stanford, he built a reputation as a dynamic researcher and educator, attracting graduate students and collaborators who were drawn to the breadth and depth of his mathematical program.

Princeton University

Sarnak joined the faculty of Princeton University, where he has held the Eugene Higgins Professor of Mathematics chair since 2002, succeeding Andrew Wiles. The Eugene Higgins Professorship is one of the most distinguished endowed chairs at Princeton, and Sarnak's appointment reflected his standing as one of the foremost mathematicians of his era.

At Princeton, Sarnak has been a central figure in the Department of Mathematics, contributing not only through his research but also through his mentorship of graduate students and postdoctoral fellows. He has supervised a large number of doctoral students, many of whom have gone on to hold positions at major research universities and institutions around the world. Sarnak also serves as an editor of the Annals of Mathematics, one of the most prestigious journals in the field, which is published by Princeton University and the Institute for Advanced Study.

Institute for Advanced Study

In 2007, Sarnak became a member of the permanent faculty of the School of Mathematics at the Institute for Advanced Study (IAS) in Princeton, New Jersey. The IAS, founded in 1930, has been the intellectual home of some of the greatest mathematicians and physicists in history, including Albert Einstein, John von Neumann, and Kurt Gödel. Membership on the permanent faculty of the School of Mathematics is one of the highest distinctions in the mathematical world, and at the time of his appointment, Sarnak joined a small group of permanent faculty members.

Sarnak holds the title of Gopal Prasad Professor of Mathematics at the Institute for Advanced Study. His dual appointment at both Princeton University and the IAS has allowed him to maintain an active research program while engaging with the broader mathematical community through the Institute's visitor programs, seminars, and workshops.

Research Contributions

Sarnak's research program is characterized by its remarkable breadth, spanning analytic number theory, automorphic forms, spectral theory, arithmetic geometry, combinatorics, and mathematical physics. His work has frequently involved establishing deep connections between seemingly disparate areas of mathematics.

His doctoral thesis on prime geodesic theorems initiated a line of research connecting the distribution of geodesics on Riemann surfaces with number-theoretic problems analogous to the prime number theorem. This spectral-geometric approach to number theory has been a recurring theme in his work and has influenced a generation of mathematicians.

One of Sarnak's notable contributions is his work on the Ramanujan conjecture and its generalizations. The Ramanujan conjecture, originally formulated by Srinivasa Ramanujan, concerns the size of Fourier coefficients of modular forms and has deep implications for number theory and the theory of automorphic forms. Sarnak has made significant progress on various aspects of this conjecture and related problems.

Sarnak has also made important contributions to the theory of expander graphs, which are sparse graphs with strong connectivity properties that have applications in computer science, coding theory, and combinatorics. According to mathematical legend, Sarnak and Noga Alon made a bet in the late 1980s about optimal graphs — specifically about the construction of graphs that achieve the best possible expansion properties. In 2025, it was reported that new research had resolved this longstanding question, proving that both Sarnak and Alon had been wrong in their respective predictions about the nature of optimal expander graphs. This episode illustrates the collaborative and sometimes playful nature of mathematical research at the highest levels, where even incorrect conjectures by leading mathematicians can stimulate decades of productive inquiry.

Sarnak's work on integral Apollonian packings — arrangements of mutually tangent circles with integer curvatures — represents another area where he has made fundamental contributions. This work connects classical geometry with modern number theory and has attracted attention from both mathematicians and mathematical popularizers. His paper on the subject received recognition from the Mathematical Association of America.

Additionally, Sarnak has contributed to the understanding of thin groups in number theory, the arithmetic of quadratic forms, quantum unique ergodicity, and the Möbius function and its connections to dynamics and complexity theory. His Möbius disjointness conjecture, which posits that the Möbius function is disjoint from any deterministic dynamical system, has become an active area of research with connections to both number theory and ergodic theory.

Mentorship and Doctoral Students

Sarnak has supervised a substantial number of doctoral students throughout his career at Stanford, Princeton, and the Institute for Advanced Study. His students have gone on to make their own contributions across a broad range of mathematical disciplines, and his influence on the development of analytic number theory and related fields extends significantly through this extensive network of academic descendants.

Collaborations and Influence

Beyond his individual research contributions, Sarnak has been a prolific collaborator. He has co-authored papers with many of the leading mathematicians of the late twentieth and early twenty-first centuries. His appreciation of the work of other mathematicians is reflected in his scholarly writing, including a 2021 essay for the Bulletin of the American Mathematical Society in which he provided an appreciation of the work of Jean Bourgain, describing how Bourgain viewed himself as an "analyst" while being "uniquely gifted as such, and much more."

In 2025, Sarnak was reported to be among the prominent mathematicians participating in Harvard University's new Benedict H. Gross Distinguished Visitors program, which aims to bring leading mathematicians to the Department of Mathematics for extended visits and collaborations.

Recognition

Sarnak has received numerous awards and honors recognizing his contributions to mathematics.

In 2024, Sarnak was awarded the Shaw Prize in Mathematical Sciences, one of the most prestigious international awards in the mathematical sciences. The announcement was made by the Shaw Prize Foundation, and Princeton University noted the award in recognition of his wide-ranging contributions. Wits University, Sarnak's undergraduate alma mater, also celebrated the award, noting his long connection with the institution.

In 2019, Sarnak received the Sylvester Medal from the Royal Society, awarded for his "transformational contributions" to mathematics. The Sylvester Medal is awarded by the Royal Society for an outstanding researcher in the field of mathematics and is named after James Joseph Sylvester.

Sarnak has been awarded the Wolf Prize in Mathematics, another of the most significant international mathematics prizes.

Sarnak is a Fellow of the Royal Society. He is also a Fellow of the American Mathematical Society, having been named among the inaugural class of AMS Fellows.

Honorary Degrees

Sarnak has received honorary degrees from multiple universities. In 2014, the University of the Witwatersrand awarded him an honorary Doctor of Science (DSc honoris causa), recognizing the achievements of one of its distinguished alumni. He has also received an honorary degree from the Hebrew University of Jerusalem. In 2015, the University of Chicago bestowed an honorary degree upon Sarnak as part of its 523rd convocation. In 2016, the University of St Andrews awarded him an Honorary Degree of Doctor of Science, with the laureation address delivered by Igor Rivin of the School of Mathematics and Statistics.

Service to the Mathematical Community

In addition to his editorial work at the Annals of Mathematics, Sarnak has served on the Board of Adjudicators and as chairman of the selection committee for the Mathematics award given under the auspices of the Shaw Prize. This role has placed him at the center of the process of recognizing outstanding mathematical achievement worldwide.

Legacy

Peter Sarnak's influence on contemporary mathematics extends across multiple dimensions. As a researcher, he has established deep and unexpected connections between number theory, geometry, combinatorics, and mathematical physics, opening new lines of inquiry that have been pursued by mathematicians around the world. His work on the Ramanujan conjecture, thin groups, expander graphs, Apollonian packings, and the Möbius disjointness conjecture has defined significant portions of the research agenda in analytic number theory and related fields during the late twentieth and early twenty-first centuries.

As a mentor, Sarnak has supervised a large number of doctoral students who have themselves become leading figures in mathematics. His academic lineage, extending from his own advisor Paul Cohen, represents a continuous tradition of mathematical innovation stretching from foundational work in set theory and logic through modern analytic number theory.

Sarnak's dual positions at Princeton University and the Institute for Advanced Study have given him a platform to influence the direction of mathematical research not only through his own work but also through his engagement with the international community of visiting scholars and collaborators who pass through these institutions. His role as an editor of the Annals of Mathematics and as an adjudicator of the Shaw Prize further amplifies his impact on the mathematical profession.

The recognition he has received — including the Shaw Prize, the Wolf Prize, the Sylvester Medal, election to the Royal Society, and multiple honorary doctorates — reflects the consensus within the mathematical community regarding the significance and breadth of his contributions. His career illustrates the power of interdisciplinary approaches within mathematics, where ideas from one field can illuminate problems in another, and where the boundaries between "pure" and "applied" mathematics are often more fluid than they might appear.