On the Greenfield-Wallach and Katok conjectures
Seminar Room 1, Newton Institute
AbstractIn 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.
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