Rudolf Zeidler: Index theory, positive scalar curvature and widths of bands

Starting with foundational work of Lichnerowicz in the 1960’s, (Atiyah-Singer) index theory has been an indispensable tool in the study of positive scalar curvature (psc) metrics. In the first part of the talk, I will give a brief introduction exposing the fundamental argument showing why index-invariants are obstructions to the existence of psc metrics.

Afterwards, we will shift gears and fast-forward to a recent geometric question due to Gromov: Let some closed manifold Mand a constant σ>0 be fixed. Suppose we are given a Riemannian metric of scalar curvature bounded below by σ on the “band” V:=M×[−1,1]. Can we find an a priori estimate on the "width of the band V"—that is, an upper bound for the distance between the two boundary components of V?

Finally, after explaining the context of this question, I will present an index-theoretic approach to such “band width” estimates.