Let G be the unit group of a quaternion algebra over a number field. The Linnik problems, solved in many cases by Duke over thirty years ago, are concerned with the equidistribution of periodic torus orbits of large discriminant on certain homogeneous spaces associated with G. Concrete examples over Q include the uniform distribution of integers points on the sphere and CM points on the modular surface. The full resolution of the Linnik problems was achieved by Michel and Venkatesh, and marked a fruitful period of exchange between ergodic theory and automorphic forms.
In the 2006 Proceedings of the ICM, Michel and Venkatesh proposed a finer way to measure the complexity of these periodic torus orbits. The ``mixing conjecture”, as it is now known, is a sort of quadratic refinement of the Linnik problems and can itself be formulated as an equidistribution statement on the product G x G.
After discussing the progression of these ideas, I will sketch a proof of the mixing conjecture, conditional on the generalized Riemann hypothesis, using techniques in analytic number theory and automorphic periods. This is joint work with Valentin Blomer and Ilya Khayutin.