  Inspired by topological data analysis techniques, we introduce persistent
homology observables and apply them in a geometric analysis of the dynamics of
quantum field theories. As a prototype application, we consider data from a
classical-statistical simulation of a two-dimensional Bose gas far from
equilibrium. We discover a continuous spectrum of dynamical scaling exponents,
which provides a refined classification of nonequilibrium self-similar
phenomena. A possible explanation of the underlying processes is provided in
terms of mixing strong wave turbulence and anomalous vortex kinetics components
in point clouds. We find that the persistent homology scaling exponents are
inherently linked to the geometry of the system, as the derivation of a packing
relation reveals. The approach opens new ways of analyzing quantum many-body
dynamics in terms of robust topological structures beyond standard field
theoretic techniques.
