**Vitali Kapovitch, University of Toronto**

**Title:** Mixed curvature almost flat manifolds

**Abstract:** A celebrated theorem of Gromov says that given n > 1
there is an ε=ε(n) > 0 such that if a closed
Riemannian manifold M^{n} satisfies the condition

-ε < sec_{M} < ε, and diam(M)< 1,
then M is diffeomorphic to an infranilmanifold. I will show
that the lower sectional curvature bound in Gromov's theorem can be
weakened to the lower Bakry-Emery Ricci curvature bound. I will also
discuss the relation of this result to the study of manifolds with
Ricci curvature bounded below.