Abstract: Taking as a starting point a question by John Lott about the vanishing of the 
-genus for spin almost-non-negatively curved manifolds, we conjecture that an almost-non-negatively curved manifold is either conformally equivalent to a manifold with positive scalar curvature or it is finitely covered by a Nilmanifold. In the way to prove such a claim, we found a generalization of Gromov’s almost flat manifold theorem where 
bounds for the curvature are relaxed to mixed curvature bounds. During the talk, we will give the precise statement of our theorem and a detailed sketch of the proof. This is a joint work with Burkhard Wilking.