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1. Background. The aim of this note is to give an explicit construction of a rich family of k-regular (except for k ° =k) of the adjacency matrix satisfy Ikjl < 2 k~-l. graphs for which all the eigenvalues kj This bound is optimal (see Proposition 2.1). We call such graphs Ramanujan graphs. These graphs have many applications in the construction of explicit… (More)

- Peter Sarnak
- 2013

Dedicated to the memory of Frances Wroblewski We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given. We also discuss the… (More)

- Peter Sarnak
- 2004

In [Ja-Sh], Jacquet and Shalika use the spectral theory of Eisenstein series to establish a new result concerning the nonvanishing of L-functions on (s) = 1. Specifically they show that the standard L-function L(s, π) of an automorphic cusp form π on GL m is nonzero for (s) = 1. We analyze this method, make it effective and also compare it with the more… (More)

Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the " function field " analogues, that is zeta functions of curves over finite fields and their generalizations. For… (More)

We review the construction of integral Apollonian circle packings. There are a number of Diophantine problems that arise in the context of such packings. We discuss some of them and describe some recent advances. 1. AN INTEGRAL PACKING. The quarter, nickel, and dime in Figure 1 are placed so that they are mutually tangent. This configuration is unique up to… (More)

- Béla Bollobás, William Fulton, Anatole Katok, Frances Kirwan, Peter Sarnak, Barry Simon +3 others
- 2015

- Jean Bourgain, Alex Gamburd, Peter Sarnak
- 2009

Dedicated to the memory of Atle Selberg

- Peter Sarnak
- 2005