Scalar curvature plays a fundamental role in geometry and in general relativity. Motivated by this connection, Misha Gromov proposed a striking rigidity conjecture: for a convex polyhedron, one cannot simultaneously increase the scalar curvature of a Riemannian metric and the mean curvature of its faces while decreasing the dihedral angles. This rigidity principle has deep consequences, including implications for the positive mass theorem in general relativity. In this talk, I will introduce Gromov’s conjecture and explain the ideas behind its proof using Dirac operators. This is joint work with Jinmin Wang and Zhizhang Xie. I will make an effort to keep the talk accessible to graduate students and non-experts.
