In this talk, we will consider Lq bounds for spectral clusters (linear combinations of eigenfunctions) of the Laplace-Beltrami operator on a compact manifold. These bounds are classical for a single cluster. We will consider N orthonormal clusters and seek bounds on the Lq norm of their square function. The main challenge is to obtain an optimal dependence on N. The results have applications to the spectral theory of Schrodinger operators and to Kakeya-type problems. The talk is based on joint work with Nhi Nguyen and Xiaoyan Su.
