On Gromov's macroscopic dimension

Gromov introduced the notion of macroscopic dimension to study closed manifolds with positive scalar curvature (PSC). In the talk we will discuss two his conjectures on the subject: I. If a closed $n$-manifold $M$ admits a PSC metric, then the macroscopic dimension of its universal cover is less than $n-1$. II. If the universal cover of a closed $n$-manifold $M$ has the macroscopic dimension less than $n$, then the image of the fundamental class under a map classifying the universal cover is trivial, $f_*([M])=0$, in the rational homology of the classifying space $H_*(B\pi)$.