Abstract: Dynamical systems on homogeneous spaces have been the subject of much attention recently, due to applications to number theory, geometric group theory, hyperbolic geometry etc. Interest in this area rose significantly after the seminal work of Margulis (the proof of the Oppenheim conjecture) and Ratner (Raghunathan’s conjectures) involving unipotent flows on homogeneous spaces. Later developments were considerably stimulated by applications to Diophantine approximation. I will survey basics of ergodic theory, then specialize to homogeneous dynamics, then highlight connections to number theory. The latter allow one to utilize certain properties of homogeneous flows, such as quantitative non-​divergence or effective equidistribution of unstable leaves, to prove new results in number theory.
Lecture 1: Introduction
Lecture 2: Homogeneous spaces, measure-preserving transformations
Lecture 3: Ergodic theorems, mixing, equidistribution
Lecture 4: Discrete subgroups and homogeneous spaces
Lecture 5: Latttices and finite volume quotients
Lecture 6: Ergodicity and mixing on homogeneous spaces
Lecture 7: Applications to number theory
Lecture 8: Exceptional orbits and number theory
Lecture 9: Shrinking targets and number theory
Lecture 10: Exponential mixing and shrinking targets
Lecture 11: Shrinking targets and Dirichlet improvability
Lecture 12: Other applications
Lecture 13: More applications and conclusion