# Laureation address: Professor Peter Sarnak

**Laureation by Professor Igor Rivin, School of Mathematics & Statistics, for Professor Peter Sarnak, recipient of the Honorary Degree of Doctor of Science**

Chancellor, it is my privilege to present for the degree of Doctor of Science, *honoris causa*, Professor Peter Sarnak.

Peter Clive Sarnak was born in Johannesburg, South Africa, in 1953. As a youth he was interested in both mathematics and chess (mostly in the latter – he was a very strong junior player), but once he started to study at the University of Witwatersrand, he became interested in mathematics to the exclusion of almost everything else. After finishing university with a Bachelor of Science and doing his compulsory military service, Peter moved to California to continue his studies with Paul Cohen at Stanford. Peter was primarily interested in logic then, but Cohen was not, and so Sarnak switched from logic to number theory. Upon finishing his PhD at Stanford in 1980, he moved to the Courant Institute (NYU) as a postdoctoral fellow. He was quickly promoted, and stayed at NYU until 1987, before returning once again to Stanford. In 1991 his final move was to Princeton, where he has remained, becoming the Eugene Higgins Professor of Mathematics in 2002 and, since 2007, he has also been a professor at the Institute for Advanced Study at Princeton.

Sarnak’s contributions to mathematics have been enormous, and while centered on number theory, have been central in such diverse fields as combinatorics, algebraic geometry, differential geometry, mathematical physics, group theory, and many others.

In his celebrated 1988 paper with Alex Lubotzky and Ralph Phillips, Sarnak constructed a family of (what they called) Ramanujan graphs – these are *expander graphs *which are, in many ways, optimal. This was, at the time, revolutionary – it was known that a random graph was an expander, but no one knew how to construct one deterministically, and certainly no one even guessed that a deterministic family with optimal expansion properties was possible. The result had immediate repercussions in theoretical computer science, but many of the techniques were developed further (often by Sarnak and his students) to create a veritable revolution in mathematics, involving such diverse fields as differential geometry and algebraic and finite groups.

In a series of papers in the late 1980s with Brad Osgood and Ralph Phillips, Sarnak developed the theory of Determinants of Laplacians (known to the mathematical physicists as Polyakov action), which provide a very powerful tool to study geometric structures on Riemann surfaces (indeed, their work resulted in a new proof of the uniformisation theorem).

In a number of papers with Nick Katz, Ze’ev Rudnick and other co-authors, Sarnak worked on the mysterious relationship between the distribution of the zeros of various (highly non-random) number-theoretically defined functions, and that of the eigenvalues of random matrices – this relationship was first noted (experimentally) by Montgomery and Odlyzko, but in the work of Sarnak and collaborators the theory took shape and put on a rigorous basis.

In recent work, Sarnak, together with collaborators, has developed the theory of thin groups, which greatly extend the range of applicability of spectral methods developed by Selberg and others.

As well as his academic career Peter is an editor of the *Annals of Mathematics *and sits on the Board of Adjudicators and the Selection Committee for the Mathematical Sciences award, given under the auspices of the Shaw Prize. He was awarded the Pólya Prize of the Society for Industrial and Applied Mathematics in 1998, the Ostrowski Prize in 2001, the Levi L Conant Prize in 2003, the Frank Nelson Cole Prize in Number Theory in 2005, and a Lester R Ford Award in 2012. He is the recipient of the 2014 Wolf Prize in Mathematics. As well as several honorary degrees from a number of universities he was elected as a member of the National Academy of Sciences and Fellow of the Royal Society in 2002.

Finally, and not surprisingly, given his range of interests, Sarnak is the guru of a very strong school of mathematics – most of his disciples are his graduate students, but some came to the fold as mature (and sometimes very prominent) mathematicians.

Chancellor, in recognition of his ground-breaking work on analytic number theory I invite you to confer the degree of Doctor of Science, *honoris causa*, on Professor Peter Sarnak.