A broad theme in geometric analysis aims to understand the geometric structure of Riemannian manifolds that satisfy constraints on curvature. The notion of scalar curvature describes how a manifold is curved on average at each point and is of fundamentalimportance in general relativity and differential geometry. We will present recent developments in the study of scalar curvature using information encoded in a family of log-Sobolev inequalities known as the Perelman entropy. This talk is based on joint workwith Man-Chun Lee and Aaron Naber.
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