Metaphors in systolic geometry

Larry Guth, University of Toronto
Fine Hall 314

The systolic inequality says that any Riemannian metric on an $n$-dimensional torus with volume 1 contains a non-contractible closed curve with length at most $C(n)$ - a constant depending only on $n$. One remarkable feature of the inequality is it holds for such a wide class of metrics. It's much more general than an inequality that holds for all metrics obeying a certain curvature condition.The systolic inequality is a difficult theorem, and each proof is guided by a metaphor that connects the systolic inequality to a different area of geometry or topology. In this talk, I will explain three metaphors. They connect the systolic inequality to minimal surface theory, topological dimension theory, and scalar curvature.