Metaphors in Systolic Geometry

Larry Guth

The systolic inequality says that if we take any metric on an n-dimensional torus with volume 1, then we can find a non-contractible curve in the torus with length at most C(n). A remarkable feature of the inequality is how general it is: it holds for all metrics. Although the statement of the inequality is short, the proofs are difficult. The general idea of each known proof comes from a metaphor connecting the systolic problem to another area of geometry/topology. I will introduce three useful metaphors, connecting the systolic problem to geometric measure theory, topological dimension theory, and scalar curvature.

https://www.ias.edu/video/members/guth