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Abstract: There are several reasons for studying flows on homogeneous spaces of Lie groups. I will discuss problems motivated by geometry of locally symmetric spaces, and by simultaneous Diophantine approximation. The results to be mentioned include: abundance of irregular geodesics, a higher rank generalization of Sullivan's logarithm law, a strengthening of Margulis' lemma on recurrence of unipotent trajectories.
By means of a correspondence between approximation properties of vectors and orbit properties of certain flows, a substantial progress has been obtained in metric number theory. The most important developments are in the field of Diophantine approximation on manifolds: new results here (joint with Margulis and Bernik) are steps towards the proof of the following vague "meta-conjecture": a Diophantine property which holds for almost all points of an ambient space should hold for almost all points on a "sufficiently curved" submanifold of the space.
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