Analysis and Probability seminar

Abstract: We consider spectral multiplier estimates for the Laplacian on flat tori. The equivalent estimates for a general compact manifold are sensitive to the curvature, by work of Huang and Sogge. One can ask just how sensitive they are, and what other data they capture. In the case of tori we conjecture the same estimates for all flat tori of a given dimension. In the limit (of small spectral window) this becomes the question of Lp norms of eigenfunctions of the Laplacian, as considered for instance by Bourgain. We discuss some results, in particular for three-dimensional tori, which use methods from number theory.

Based on joint work with Pierre Germain and Daniel Pezzi.