Metaphors in systolic geometry

TL;DR

This paper provides an expository overview of systolic geometry, highlighting key theorems, the challenges in their proofs, and various metaphors linking the subject to other geometric concepts.

Contribution

It introduces diverse metaphors that offer new perspectives and approaches to understanding central problems in systolic geometry.

Findings

Connections between systolic inequalities and minimal surfaces

Metaphors linking systolic geometry to hyperbolic geometry and scalar curvature

Insights into the complexity of proofs in systolic geometry

Abstract

This is an expository essay about systolic geometry. It describes a central theorem in the subject and why the proof is difficult. Then it discusses different metaphors which suggest ways to approach the problem. The metaphors connect the systolic inequality to minimal surfaces, topological dimension, scalar curvature, and hyperbolic geometry.