Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures

摘要

The discussion starts with a macroscopic view on Riemannian manifolds withpositive scalar curvature and terminates with a glimpse on the proof of the homotopy invariance of some Novikov's higher signatures of non-simply connected manifolds. The approach focuses on the spectra of geometric differential operators on compact and non-compact manifolds V where the link with the macroscopic geometry and topology is established with suitable index theorems for the operators twisted with almost flat bundles over V. The perspective comes mainly from the asymptotic geometry of infinite groups and foliations.