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This paper is a slightly enlarged version of a series of lectures on the Linnik problems given at the SMS–NATO ASI 2005 Summer School on Equidistribution in Number Theory.

Measure rigidity is a branch of ergodic theory that has recently contributed to the solution of some fundamental problems in number theory and mathematical physics. Examples are proofs of quantitative versions of the Oppenheim conjecture (Eskin et al., 1998), related questions on the spacings between the values of quadratic forms (Eskin et al., 2005; Marklof, 2003; Marklof, 2002), a proof of quantum unique ergodicity for certain classes of hyperbolic surfaces (Lindenstrauss, 2006), and an approach to the Littlewood conjecture on the nonexistence of multiplicatively badly approximable numbers (Einsiedler et al., 2006).

These are the notes to accompany some lectures delivered at the 2005 NATO ASI summer school in Montréal. They constitute an introduction to the spectral theory of automorphic forms. The viewpoint is slightly nonstandard, in that we present first the “group representation” viewpoint and later descend to the upper-half plane.

We give examples of how classifying invariant probability measures for specific algebraic actions can be used to prove density and equidistribution results in number theory.

These notes contain crash courses on classical and quantum mechanics and on semi-classical analysis as well as a short introduction to one issue in quantum chaos: the semi-classical eigenfunction behaviour for quantum systems having an ergodic classical limit. The emphasis is on explaining the conceptual and structural similarities between the ways in which this question arises in the study of arithmetic surfaces and ergodic toral automorphisms. The text is aimed at an audience of graduate students and post-docs in number theory.

In these lectures I describe in detail the quantization of linear symplectic maps of the torus, as a continuation of De Bievre’s lectures (De Bi`evre, 2006). I will then survey the problem of quantum equidistribution for this model. This model was introduced by Hannay and Berry (Hannay and Berry, 1980). It turns out that it has a rich arithmetic structure, and its study uses several ingredients in modern number theory.