Abstract: Rigidity properties of diagonalizable flows on homogeneous spaces

Actions of multiparameter diagonalizable group on homogeneous spaces, such as the action of the group of $d \times d$-diagonal matrices on the space of unit volume lattices in $R^d$ for $d\geq 3$, display subtle rigidity properties that are only very partially understood. This partial understanding, however, is already sufficient for many applications, in particular in number theory.
I will present these rigidity properties, and compare them to the better understood rigidity properties of unipotent flows. Much of the progress understanding these group actions goes via the study of invariant measures, and I will present some of the results in this direction.

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