### On the Greenfield-Wallach and Katok conjectures

Flaminio, L *(Université Lille 1)*

Tuesday 01 July 2014, 13:30-14:20

Seminar Room 1, Newton Institute

#### Abstract

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.

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