I will present quantitative equidistribution results for the action of Abelian subgroups of the (2g+1)-dimensional Heinberg group on a compact homogeneous nilmanifold. The results are based on the study of the cohomology of the action of such groups on the algebra of smooth functions on the nilmanifold, on time estimates of the associated cohomological equations with respect to a suitable Sobolev grading, and on renormalization in an appropriate moduli space (a method applied by Forni to surface flows and by Forni and Flaminio to other parabolic flows). As an application, we obtain bounds on finite Theta sums defined by real quadratic forms in g variables, depending on certain Diophantine conditions, generalizing to higher dimension the classical bounds by Hardy and Littlewood (1914) and by Fiedler, Jurkat and Korner (1977). This is joint work with Livio Flaminio, http://arxiv.org/abs/1504.06795
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