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On the Greenfield-Wallach and Katok conjectures

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If you have a question about this talk, please contact Mustapha Amrani .

Interactions between Dynamics of Group Actions and Number Theory

In the early 70’s, Greenfield and Wallach studied fields globally hypo-elliptic vectors fields on compact manifolds and made the following conjecture :

``Let $ G $ be a Lie group and let $ H $ be a closed subgroup Such That $G/H
$ is compact.  Let $ X   $ be the vector field on $ G / H $ determined by
some element $X$ in the Lie algebra of $G$. Given by If $ X   $ is globally
hypo-elliptic, then $ G / H $ is a torus.''
In a work in collaboration with F.~Rodriguez-Hertz and G. Forni,
we gave a positive solution to this problem.
In this talk I will recall  thistory of the problem, wxplain its relation with Katok's conjecture
on cohomologically free vector fields and give some idea of the proof.

This talk is part of the Isaac Newton Institute Seminar Series series.

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