There was another girl in the class and she showed me how to solve two simultaneous linear equations and I was just blown away about this and I just thought she was the greatest genius and I was just following her around. I asked her to explain something else to me - so she showed me how to eliminate variables which essentially is a big chunk of linear algebra. This was fascinating to me and then I showed it to my father and he knew how to do that too.

... my passion was tournament chess where I enjoyed success at the junior and senior levels in southern Africa. My father was very supportive of my (and my brothers') involvement with chess while we were kids ...

... when I was playing chess, I would read all the chess books from the local library of all the grandmasters. One of the grandmasters was a Russian Botvinnik, world champion around 1960. He had a degree in electrical engineering and would write in some of his books sort of mathematically. ... engineers used the letter j not i for the square root of minus one. ... he said it's the square root of minus one and I learned that from a world chess champion, and well, I understood what he was saying but he was using it as an analogy.

What won me over as a first year undergraduate studying mathematics was abstraction and specifically that conceptual thinking can make the solution of a problem and understanding of a theory completely transparent. I remember the first course in abstract linear algebra as a spark, and also a topology course that drew me to want to learn and understand much more.

... completed his BSc degree at the University of the Witwatersrand in 1974. He was awarded the Herbert Le May prize for Applied Mathematics and the William Cullen Medal for the best graduate in the Faculty of Science in 1974. He went on to complete his BSc Honours in Pure Mathematics also at the University of the Witwatersrand and was awarded the Unico Chemical Company Gold Medal for the Best Honours student in the Science Faculty in 1975.

One of my teachers D B Sears (who was one of the few students of the British analyst E C Titchmarsh and worked on spectral theory of differential operators) gave a course on Gelfand's theory of Banach Algebras. The course was aimed at showing the power of this elegant theory and its broad applications, and it left a tremendous impression on me - it was the first piece of modern mathematics that I had seen.

I had taken some basic courses in mathematical logic and even some about Cohen's technique of forcing. I had heard from some of the faculty that Cohen was a very dynamic and brilliant mathematician and found this very appealing. This information was rather accurate and I was very fortunate to learn a great amount of mathematics and especially taste and quality from Paul Cohen.

Paul Cohen, who was my thesis advisor, had a major influence on my mathematical taste, knowledge, insight and intuition. His view of the unity of mathematics (and that one really need not stick to a small sub-field) made a big impression on me. Together, Paul and I studied a good portion of Selberg's works, and the core of my own work is very much shaped and influenced by Selberg's ideas.

I would like to thank my adviser, Professor P Cohen, for all he taught me over the past few years, and for his encouragement and advice. I would also like to thank Professor R S Phillips for his encouragement and the many discussions on and relating to the subject of this thesis. To my office mate, A Woo, thanks for all the mathematics learned and done together. ... Finally, thanks to Helen for her continual support.

From the point of view of students and young postdocs, Ralph was a model professor. He always welcomed people into his office to discuss mathematics or to do joint readings. ... In seminars and colloquia he never hesitated to ask a basic question, even though it might well show some ignorance. His were the questions that many in the audience were wondering about but were afraid to ask. In fact, Ralph was never interested in dazzling; he never put on airs. He was happy and satisfied with what he was doing, and it was contagious. He much preferred to work together with others, and as is clear from his many successful collaborations, he was very good at it.

A lot of work has been done in recent years concerning closed geodesics on Riemannian manifolds. In the case that the manifold has a "large" fundamental group such as a surface of genus g ≥ 2, the existence of infinitely many such geodesics is easily demonstrated by exhibiting one in each free homotopy class. The work referred to above is concerned with showing the existence of infinitely many prime geodesics on any compact, closed manifold.

It is our aim in this thesis to study the asymptotic distribution of the lengths of periodic geodesics, and consequences thereof. The results we obtain have interesting applications to number theory as well as to the behaviour of eigenvalues of the Laplace-Beltrami operator on Riemannian manifolds.

There are three different methods of getting hold of the closed geodesics. The first is to use the fact that these geodesics are critical points of the energy integral on a path space. Secondly, by viewing these geodesics as periodic orbits of the geodesic flow, one may use methods of topological dynamics. Finally, in the case of constant curvature, the lengths of the closed geodesics can be identified through exact trace formulas such as the Selberg Trace Formula. We will be concerned mainly with the latter two methods, as they are more useful as far as asymptotics of the lengths is concerned.

The recipients were selected on the basis of their exceptional potential to make creative contributions to scientific knowledge. ... Candidates for fellowships are nominated by senior scientists familiar with their talents. Fellows need not pursue a specified research project and are free to shift the direction of their research at any time.

The National Science Foundation recently announced ... the Presidential Young Investigator Awards for 1985. This is the second year in a program begun in 1984. ... The awards, which fund research by faculty near the beginning of their academic careers, are intended to help universities attract and retain outstanding young Ph.D.'s who might otherwise pursue non-teaching careers.
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Names of 1985 recipients of Presidential Young Investigator Awards in the mathematical sciences, their institutional affiliations and research interest follow: ... Peter Sarnak (Stanford University), Analysis, Number Theory and Geometry ...

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 this lecture we describe and exploit the relation between analytic Diophantine problems on homogeneous varieties and harmonic analysis on the corresponding groups. For the case G = SL(2) this relation has been well studied and striking applications to analytic number theory have been found, especially by Iwaniec and his collaborators. Our focus here will be on general G where this aspect of the theory is still in a primitive state.

Since Hecke's work, the theories of L-functions and of automorphic forms have been closely interwoven. In this talk, we review some recent developments concerning the analytic aspects of these topics.

I have worked in areas ranging from analysis to number theory and mathematical physics. A recurring theme throughout this work is the role of symmetry and group theory. The modern theory of zeta and L-functions, which has its origins in the works of Dirichlet and Riemann, has far-reaching applications to prime numbers, to Diophantine equations such as the solution of quadratic equations in several integer variables, to combinatorics and theoretical computer science, and even to the understanding of the quantizations of certain arithmetically defined chaotic Hamiltonian systems. Finding and exploiting these applications in order to solve basic problems of these types has been one of the major thrusts of my work.

... pathbreaking extension of steepest descent methods for the asymptotic analysis of oscillatory Riemann-Hilbert problems.

His contributions to number theory and to questions of analysis often motivated by number theory have been very influential in mathematics.

... expository paper "Zeroes of zeta functions and symmetry". [It] is a model of high-level exposition. Katz and Sarnak do justice to their beautiful topic, a rich mix of intensive numerical exploration, conjectures, and theorems.

... for his work relating the distribution of zeros of L-functions in certain families to the distribution of eigenvalues in a large compact linear group of a type that depends on the family of L-functions one is considering.

... for his deep contributions to analysis, number theory, geometry, and combinatorics.

... for transformational contributions across number theory, combinatorics, analysis and geometry.

... this honour is awarded based on skill in instruction, commitment to working with and building relationships with undergraduates, and ability to spark students' interests. To this end, senior Daniel Kriz had this to say about Sarnak: "Perhaps my favourite undergraduate maths class at Princeton was MAT 415 Analytic Number Theory. … [Professor Sarnak] always paid careful attention to make sure the entire class was up to speed during lecture, and the assigned problem sets which he personally devised were not only illuminating and thought-provoking, but (dare I say) fun as well. Moreover, he was more than willing to go over material both after lecture and outside the classroom; I have not met another instructor so happy to meet outside of class, sit down and point to exactly where, how and why my proof was wrong."

By far my biggest mentor was Peter Sarnak, my advisor in graduate school. Grad school was (for me and for many others) a time of real struggle. Before that it was all "learn this subject, do some problems sets, learn some more, you're doing great!" but in Grad school there's a need to produce original work, which I had never done or even considered really until that point. Peter is extraordinarily supportive, enthusiastic, and energetic. Every time I met with him I left feeling like a million bucks. This is pretty rare in an advisor and I really needed it. On a more technical level a lot of my interests in mathematics were taken in one form or another from Peter. He was (and still is) very good at picking problems for me to work on and the directions he pushed me in have proven fruitful.

Sarnak is a very positive person. You would go in to see him and, of course, you'd be depressed because you hadn't been able to do anything. But you'd come out of his office feeling cheerful. You'd come out feeling you're working on the right thing. I think that's actually very valuable - that he was able to kind of make you feel positive. He was able to motivate you. That was really valuable for me. For a graduate student it can be very difficult. It's very easy to get discouraged if you have no real understanding that you're going to be stuck for a very long time.

This book treats in detail a remarkable range of ideas and beautiful mathematics. It is highly recommended to everyone interested in modular forms.

This book is fascinating in many aspects: First, its rigorous, systematic and accessible exposition of the subject makes it a bright landmark at the crossroads of arithmetic and mathematical physics; no doubt it will become a basic reference in random matrix theory. Second, it offers its reader a bouquet of beautiful new results but also leaves the door open to many challenging conjectures.

It would make a great text for an honours or senior seminar, showing how elegantly many different areas of mathematics come together to solve a very concrete problem of broad interest and application.

Leading number theorist Peter Sarnak has joined the Faculty of the School of Mathematics at the Institute for Advanced Study. Sarnak, a Member at the Institute from 1999 to 2002 and from 2005 to 2007, is the Eugene Higgins Professor of Mathematics at Princeton University. He will continue to hold his appointment at Princeton in conjunction with his professorship at the Institute. Peter Goddard, Director of the Institute, described Sarnak as a scholar who "combines distinction as one of the world's leading number theorists with outstanding talents as a mentor of younger mathematicians. He inspires enormous enthusiasm for his subject and fosters interactions across disciplinary boundaries."
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Regarding his appointment, Sarnak commented, "I am delighted to join the Institute for Advanced Study and to participate in its primary mission of scholarship, research, and mentoring. To step into the unique intellectual environment that has defined the Institute since its beginnings is a challenge and an opportunity that I am grateful to embrace."

The Institute is proud to announce the creation of the Gopal Prasad Professorship in recognition of prolific mathematician and six-time Member of Institute for Advanced Study (IAS), Gopal Prasad. The professorship, endowed with a gift from the Prasad family, ensures that future generations of scholars, from all regions of the world, have the opportunity to benefit from the unique environment of discovery at IAS. Peter Sarnak, current Professor in the School of Mathematics, has been selected as the inaugural Gopal Prasad Professor. The professorship is to be held by Faculty in the Schools of Mathematics and Natural Sciences.