Quantitative Constraints under Positive Scalar Curvature

Recent developments in scalar curvature geometry have revealed that positive scalar curvature (PSC) imposes quantitative restrictions on various geometric invariants. In this talk, I will present two results in this direction: an estimate for the stable two-dimensional systole of compact PSC manifolds via Dirac operator methods, and a decay estimate for PSC on a class of noncompact four-manifolds. I will first review applications of the index theorem to positive scalar curvature due to Gromov and Lawson, together with geometric measure theoretic approaches to systolic inequalities. I will then explain the main ideas behind the proofs of the above results.