Some counting problems admit extra symmetries, such as invariance under actions of a semisimple Lie group. These problems can sometimes be solved by techniques involving ergodic theory. We are working on examples from number theory, geometry and dynamics. The phenomenon of symmetry is central to certain counting problems. Typically these problems cannot be solved unless all of the hidden symmetry is exploited. The proposed methods consist of studying the interplay between the symmetric patterns and the chaotic phemomena of ergodic theory.