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Abstract

Many non-perturbative phenomena in quantum field theories are driven or accompanied by non-local excitations, whose dynamical effects can be intricate but difficult to study. Amongst others, this includes diverse phases of matter, anomalous chiral behavior, and non-equilibrium phenomena such as non-thermal fixed points and thermalization. Topological data analysis can provide non-local order parameters sensitive to numerous such collective effects, giving access to the topology of a hierarchy of complexes constructed from given data. This dissertation contributes to the study of topological data analysis and geometry in quantum field dynamics. A first part is devoted to far-from-equilibrium time evolutions and the thermalization of quantum many-body systems. We discuss the observation of dynamical condensation and thermalization of an easy-plane ferromagnet in a spinor Bose gas, which goes along with the build-up of long-range order and superfluidity. In real-time simulations of an over-occupied gluonic plasma we show that observables based on persistent homology provide versatile probes for universal dynamics off equilibrium. Related mathematical effects such as a packing relation between the occurring persistent homology scaling exponents are proven in a probabilistic setting. In a second part, non-Abelian features of gauge theories are studied via topological data analysis and geometry. The structure of confining and deconfining phases in non-Abelian lattice gauge theory is investigated using persistent homology, which allows for a comprehensive picture of confinement. More fundamentally, four-dimensional space-time geometries are considered within real projective geometry, to which canonical quantum field theory constructions can be extended. This leads to a derivation of much of the particle content of the Standard Model. The works discussed in this dissertation provide a step towards a geometric understanding of non-perturbative phenomena in quantum field theories, and showcase the promising versatility of topological data analysis for statistical and quantum physics studies.