I will start this talk with a brief introduction to Quantum Field Theory and motivate the occurring Feynman integrals. These integrals, which can be interpreted as a perturbative definition of the path integral, are typically divergent and thus ill-defined. To remedy this, renormalization theory has been introduced, which I will explain in the formulation of Connes and Kreimer: The first part is to set up a Hopf algebra of Feynman graphs, which then allows to construct renormalized Feynman rules via an algebraic Birkhoff decomposition in the respective character group. Finally, I want to give an outlook how I aim to apply this framework to gauge theories and gravity: The first step is to manifest (generalized) Slavnov—Taylor identities via a Hopf ideal, a result originally due to van Suijlekom and refined by myself. Then, as work in progress, I aim to encode cancellation identities via an appropriate Feynman graph cohomology. This then combines to a differential-graded quotient renormalization Hopf algebra, whose relation to the BV-BRST formalism will be studied in future work. More information can be found in my dissertation and the articles it is based upon, cf. arXiv:2210.17510 [hep-th].

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