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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.