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 two useful metaphors, connecting the systolic problem to topological dimension theory and scalar curvature.