Mathematical Conversations

Abstract: In 2010, Larry Guth wrote a beautiful essay "Metaphors in systolic geometry", where he poetically described several approaches to Gromov's celebrated systolic inequality. A nontrivial special case of this inequality claims that a riemannian n-torus always has a non-contractible loop of length at most the n-th root of the volume. One of the metaphors draws the intuition from geometry of scalar curvature, namely, the Schoen--Yau method of proving that the n-torus does not admit a metric of positive scalar curvature. I will describe the metaphor and survey some of its recent applications.