i will survey recent results regarding the dynamics of positive definite functions and character rigidity of irreducible lattices in higher-rank semisimple algebraic groups. these results have several applications to ergodic theory, topological dynamics, unitary representation theory, and operator algebras. in the case of lattices in higher-rank simple algebraic groups, i will explain the key operator algebraic novelty, which is a noncommutative nevo-zimmer theorem for actions on von neumann algebras. i will also present a noncommutative margulis' factor theorem and discuss its relevance regarding connes' rigidity conjecture for group von neumann algebras of higher-rank lattices.