The Mixing Conjecture of Michel-Venkatesh has now taken on additional arithmetic significance via Wiles' new approach to modularity. Inspired by this, we present the best currently available method, pioneered by Khayutin's proof for quaternion algebras over the rationals, which we have successfully applied to totally real fields. The talk will overview the method, which brings a suprising combination of ergodic theory, analysis and geometry to bear on this arithmetic problem.

We study fully discrete approximation of the 2D Euler equations for ideal homogeneous fluids. We focus on spectral methods and  discuss rates of convergence of velocity and vorticity under different assumptions on the smoothness of the data.