Scalar Curvature Rigidity of Polytopes

A central theme in differential geometry involves studying Riemannian metrics satisfying various curvature positivity conditions. The weakest condition one can impose is the positivity of the scalar curvature. Inspired by Toponogov's triangle comparison theorem on manifolds with nonnegative sectional curvature, Gromov has proposed a comparison principle for polytopes carrying metrics with nonnegative scalar curvature. In this lecture, we will discuss a recent result which verifies Gromov's conjecture under a stronger hypothesis. The proof uses the Fredholm theory for Dirac operators on manifolds with boundary, as well as an estimate from harmonic analysis due to Fefferman and Phong.