Over the past three decades, the ergodic theory for flows on homogeneous spaces has produced many beautiful applications in number theory, geometry and mathematical physics. In these lectures I will provide an introduction to equidistribution theorems for large spheres and horospheres, and explain their very recent application to directional statistics for lattices, quasicrystals and other aperiodic point sets. These lectures are introductory and aimed at a non-specialist mathematical audience. Much of the material is based on joint work with A. Strombergsson, D. El-Baz and I. Vinogradov.

Over the past three decades, the ergodic theory for flows on homogeneous spaces has produced many beautiful applications in number theory, geometry and mathematical physics. In these lectures I will provide an introduction to equidistribution theorems for large spheres and horospheres, and explain their very recent application to directional statistics for lattices, quasicrystals and other aperiodic point sets. These lectures are introductory and aimed at a non-specialist mathematical audience. Much of the material is based on joint work with A. Strombergsson, D. El-Baz and I. Vinogradov.