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Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. (English) Zbl 1161.15302
Summary: Typical large unitary matrices show remarkable patterns in their eigenvalue distribution. These same patterns appear in telephone encryption, the zeros of Riemann’s zeta function, a variety of physics problems, and in the study of Toeplitz operators. This paper surveys these applications and what is currently known about the patterns.

MSC:
15A18 Eigenvalues, singular values, and eigenvectors
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
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