From The Institute for Advanced Study and Princeton University ,Princeton,

NJ .USA

**Prof. Sarnak** is a mathematician of an extremely broad spectrum with a far-reaching vision. He has impacted the development of several mathematical fields, often by uncovering deep and unsuspected connections. In analysis, he investigated eigenfunctions of quantum mechanical Hamiltonians which correspond to chaotic classical dynamical systems in a series of fundamental papers. He formulated and supported the “Quantum Unique Ergodicity Conjecture” asserting that all eigenfunctions of the Laplacian on negatively curved manifolds are uniformly distributed in phase space. Sarnak's introduction of tools from number theory into this domain allowed him to

obtain results which had seemed out of reach and paved the way for much further progress, in particular the recent works of E. Lindenstrauss and N. Anantharaman. In his work on L-functions (jointly with Z. Rudnick) the relationship of contemporary research on automorphic forms to random matrix theory and the Riemann hypothesis is brought to a new level by the computation of higher correlation functions of the Riemann zeros. This is a major step forward in the exploration of the link between random matrix theory and the statistical properties of zeros of the Riemann zeta function going back to H. Montgomery and A. Odlyzko. In 1999 it culminates in the fundamental work, jointly with N. Katz, on the statistical properties of low-lying zeros of families of L-functions. Sarnak’s work (with A. Lubotzky and R. Philips) on Ramanujan graphs had a huge impact on combinatorics and computer science. Here again he used deep results in number theory to make surprising and important advances in another discipline.

By his insights and his readiness to share ideas he has inspired the work of students and fellow researchers in many areas of mathematics.