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The theory of elementary catastrophes has inspired a new non-Euclidean quantum mechanics-based geometry to describe the topology of molecules and clusters. The Perspective article by Samantha Jenkins on page 1603 discusses the current research that explores the relationship between the topology of the molecules and the phase space for both the molecular and the solid-state topologies, as well as an alternative application of the Poincaré-Hopf relation. The four different types of quantum geometry dimension are shown in the cover image along with the corresponding form of the Poincaré-Hopf relation depending on the types of critical point present.