Renormalisation under the prism of meromorphic functions in several variables

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Meromorphic functions are central tools in the context of renormalisation and meromorphic functions in several variables with linear poles naturally arise from a multiparameter renormalisation. They are ubiquous in mathematics and physics, where they appear in various disguises such as Feynman integrals, discrete sums on cones and multizeta functions. Thus their study touches on various areas of mathematics, including number theory, toric geometry, category theory and infinite-dimensional topology. When compared with the more commonly used one parameter renormalisation, a multiparameter renormalisation operates a shift from the study of the (co-) algebraic structure underlying the subdivergences to be renormalised to the study of the pole structure of the resulting meromorphic germs. Hence, algebras of meromorphic functions in several variables provide a common tool to handle various renormalisation problems. In this framework, the renormalisation takes place in form of a generalised evaluator, a linear form on an algebra of meromorphic functions, which extends the ordinary evaluation at a point to an appropriate evaluation at poles. A classification of the generalised evaluators thanks to a group playing the role of a renormalisation group is the first main objective of this project. Such a classification can be carried out thanks to an underlying locality relation, a symmetric binary relation on meromorphic germs which has a separating role, serving the purpose of separating the subdivergences to be renormalised. Matching the needs of a multiparameter renormalisation requires enhancing the algebraic locality framework developed in previous work to a topological locality setup as well as generalising the related analytic tools beyond the orthogonality binary relation, which is the second challenging objective in this project. Our third aim is the study of the pole structure of regularised amplitudes derived from graphs decorated by holomorphic families of classical pseudodifferential operators and to capture their pole structure by universal properties of graphs using PROPs and their generalisations.