Prof. Mikhail Gromov | 100 Problems around Scalar Curvature

In this lecture, Mikhail Gromov presents a collection of open problems and conjectures centered around scalar curvature. He begins by discussing geometric deformations of simplices and polyhedra, asking whether one can deform faces to have positive mean curvature while decreasing dihedral angles. He then moves to questions about the curvature of submanifolds in Euclidean space, relating topology to curvature bounds, and mentions connections to K-theory and the index theorem. Gromov introduces a conceptual definition of scalar curvature based on volume comparison and multiplicativity, and discusses a theorem about cubes: under positive scalar curvature and mean convex faces with angles less than pi/2, the cube must be Euclidean. He sketches a proof using reflection to construct a torus with positive scalar curvature, leading to a contradiction via the index theorem and a Bochner-type argument. He also touches on the role of spin structures and the challenges of extending these results to non-spin manifolds. Throughout, he emphasizes the need for a deeper understanding of the fundamental nature of manifolds and scalar curvature.