Diophantine Problems and Homogeneous Dynamics

We will discuss some classical questions that have their origins in the work of Gauss from 1863 [16, pp. 269–291], along with some more recent developments due primarily to Duke, Rudnick and Sarnak [12], Eskin and McMullen [15]. Rather than aiming for maximal generality, we will try to expose the striking connection between certain equidistribution problems in dynamics and some asymptotic counting problems. In the second part we will discuss the connection between the theory of Diophantine approximation and homogeneous dynamics. In these notes we will make use of several concepts from ergodic theory and dynamical systems. For most of these a suitable source is [13], and for some of the material on the space of lattices particularly a suitable source is [14].