mathematics

Peter Sarnak

Peter Sarnak is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Peter Sarnak rather than just read about it. In short: Peter Clive Sarnak (born 18 December 1953) is a South African and American mathematician. Sarnak has been a member of the permanent faculty of the School of Mathematics at the Institute for Advanced Study since 2007.

Key takeaways

Peter Sarnak belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Peter Sarnak to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Peter Sarnak from memory before moving on to harder problems.

Reference excerpt

Peter Clive Sarnak (born 18 December 1953) is a South African and American mathematician. Sarnak has been a member of the permanent faculty of the School of Mathematics at the Institute for Advanced Study since 2007. He has also been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Sir Andrew Wiles, and is an editor of the Annals of Mathematics. He is known for his work in analytic number theory. He was member of the Board of Adjudicators and for one period chairman of the selection committee for the Mathematics award, given under the auspices of the Shaw Prize.

Education Sarnak is the grandson of one of Johannesburg's rabbis and lived in Israel for three years as a child. He graduated from the University of the Witwatersrand (BSc 1975, BSc(Hons) 1976) and Stanford University (PhD 1980), under the direction of Paul Cohen. Sarnak's work (with A. Lubotzky and R. Phillips) applied results in number theory to Ramanujan graphs, with connections to combinatorics and computer science.

Career and research Sarnak has made contributions to analysis and number theory. He is recognised as one of the leading analytic number theorists of his generation. His early work on the existence of cusp forms led to the disproof of a conjecture of Atle Selberg. He has obtained the strongest known bounds towards the Ramanujan–Petersson conjectures for sparse graphs, and was one of the first to exploit connections between certain questions of theoretical physics and analytic number theory. There are fundamental contributions to arithmetical quantum chaos, a term which he introduced, and to the relationship between random matrix theory and the zeros of L-functions. His work on subconvexity for Rankin–Selberg L-functions led to the resolution of Hilbert's eleventh problem. During his career he has held numerous appointments including:

Assistant Professor, 1980–83; Associate Professor, 1983; Professor, 2001–2005, Courant Institute, New York University Associate Professor, 1984–87; Professor, 1987–91, Stanford University Professor, 1991–; H. Fine Professor, 1995–96; Chairman, Dept of Mathematics, 1996–99; Eugene Higgins Professor, 2002–, Princeton University Member, 1999–2002 and 2005–2007; Faculty, 2007–present, Institute for Advanced Study

Publications Sarnak, P. (1982). "Spectral Behavior of Quasi Periodic Potentials". Commun. Math. Phys. 84 (3): 377–401. Bibcode:1982CMaPh..84..377S. doi:10.1007/bf01208483. S2CID 123319103. Some Applications of Modular Forms, 1990 (joint editor) Extremal Riemann Surfaces, 1997 (joint author) Random Matrices, Frobenius Eigenvalues and Monodromy, 1998 Peter Sarnak (2000). "Some problems in Number Theory, Analysis and Mathematical Physics". In V. I. Arnold; M. Atiyah; P. Lax; B. Mazur (eds.). Mathematics: frontiers and perspectives. American Mathematical Society. pp. 261–269. ISBN 978-0-8218-2697-3. (joint editor) Selected Works of Ilya Piatetski-Shapiro (Collected Works), 2000 (joint author) Elementary Number Theory, Group Theory and Ramanujan Graphs, 2003 (joint editor) Selected Papers Volume I-Peter Lax, 2005 (joint editor) Automorphic Forms and Applications, 2007

Awards and honours Peter Sarnak was awarded:

1988 Pólya Prize of the Society for Industrial & Applied Mathematics 2001 Ostrowski Prize 2003 Levi L. Conant Prize 2005 Frank Nelson Cole Prize in Number Theory 2012 Lester R. Ford Award 2014 Wolf Prize in Mathematics 2019 Sylvester Medal of the Royal Society 2024 Shaw Prize. The University of the Witwatersrand conferred an honorary doctorate on Professor Peter Sarnak on 2 July 2014 for his distinguished contribution to the field of mathematics. He was an invited speaker at the International Congress of Mathematicians (ICM) in 1990 in Kyoto and a plenary speaker at the ICM in 1998 in Berlin. Sarnak has been a member of the American Academy of Arts and Sciences since 1991. He was also elected as member of the National Academy of Sciences (USA) and Fellow of the Royal Society (FRS) in 2002. He became a member of the American Philosophical Society in 2008. He was awarded an honorary doctorate by the Hebrew University of Jerusalem in 2010. He was also awarded an honorary doctorate by the University of Chicago in 2015 and by Stockholm University in 2023. He was elected to the 2018 class of fellows of the American Mathematical Society. In 2019 he became the 10th non-British citizen to ever be awarded the Sylvester Medal of the Royal Society.

References

This article incorporates text available under the CC BY 4.0 license.

Illustrations

Worked examples

Example 1 — a first encounter with Peter Sarnak

Start with the simplest possible case. Write down what Peter Sarnak claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Peter Sarnak before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Peter Sarnak ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Peter Sarnak

In research: Peter Sarnak appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Peter Sarnak in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Peter Sarnak is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1953 births, 20th-century American mathematicians, 21st-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Peter Sarnak outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Peter Sarnak in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Peter Sarnak means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Peter Sarnak out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Peter Sarnak in simple terms?

Why does Peter Sarnak matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Peter Sarnak?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Peter Sarnak.