The non-Euclidean lattice points counting problem

Euclidean lattice points counting problems, the primordial example of which is the Gauss circle problem, are an important topic in classical analysis. Their non-Euclidean analogs in irreducible symmetric spaces (such as hyperbolic spaces and the space of positive-definite symmetric matrices) are equally significant, and we will present an approach to establishing such results in considerable generality. Our method is based on dynamical arguments together with representation theory and non-commutative harmonic analysis, and produces the current best error estimate in the higher rank case. We will describe some of the remarkably diverse applications of lattice point counting problems, as time permits.