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Monday, November 10, 2014

**Abstract:** The Linearization Theorem for Lie groupoids provides an organizing framework for classic results on the geometry of fibrations, actions and foliations. It was conjectured by A. Weinstein, who also gave a Morita invariance argument reducing the problem to the fixed point case, later solved by N. Zung. In a joint work with R. Fernandes we approach the linearization problem from a new perspective, developing a notion of metrics on Lie groupoids, achieving a simpler proof and a stronger theorem. Morita invariance is not needed in our approach, but a version of it still holds, leading to a definition of metrics on stacks. I will recall the interplay between groupoids and stacks, discuss the theory of Riemannian groupoids, and present some of our next results.