Speaker: Balarka Sen
Date: February 05, 2026
Affiliation: TIFR, Mumbai

Abstract: Scalar curvature of a Riemannian manifold at a point is, upto a normalizing constant, the average sectional curvature of all $2$-dimensional planes passing through that point. An outstanding conjecture by Gromov posits that every complete Riemannian manifold with scalar curvature bounded below by a uniform positive constant, looks (at a certain scale) collapsed to a simplicial complex of codimension at least two. Positivity of the scalar curvature implies that the volume of any geodesic ball of infinitesimal radius in the manifold is less than that of a Euclidean ball of comparable radius. Therefore, one might hope to prove a version of Gromov's conjecture, replacing a scalar curvature bound by the following ``positive macroscopic scalar curvature" condition: volume of any unit ball inside the universal cover of a concentric ball of radius $2$, is bounded above by a uniform small constant. Guth (2011, Annals of Math.) showed that in this case, the manifold looks collapsed to a codimension $1$ simplicial complex. We show that the macroscopic version of Gromov's conjecture is false in dimensions four and above: there exists Riemannian manifolds which do not look collapsed in two or more directions, but nevertheless have arbitrarily large positive macroscopic scalar curvature. This is a joint work with Aditya Kumar (University of Maryland, College Park).