Renormalisation under the prism of meromorphic functions in several variables

亚纯函数是重整化背景下的核心工具，具有线性极点的多变量亚纯函数自然产生于多参数重整化。它们在数学和物理中无处不在，它们以各种形式出现，如费曼积分，锥上的离散和和多ζ函数。因此，他们的研究涉及数学的各个领域，包括数论、环面几何、范畴论和无限维拓扑。与更常用的单参数重整化相比，多参数重整化操作从研究要重整化的子散度的（协）代数结构转向研究由此产生的亚纯胚的极点结构。因此，多变量亚纯函数的代数提供了一个处理各种重整化问题的通用工具。在这个框架中，重整化以广义求值器的形式发生，这是亚纯函数代数上的线性形式，它将点上的普通求值扩展到极点上的适当求值。这个项目的第一个主要目标是通过一个扮演重整化小组角色的小组对广义评估者进行分类。这样的分类可以通过一个潜在的局部性关系来实现，这是一个亚纯胚上的对称二元关系，它具有分离作用，用于分离待重整化的子散度。匹配多参数重整化的需求需要将以前工作中开发的代数局部性框架增强到拓扑局部性设置，并将相关分析工具推广到正交二元关系之外，这是本项目的第二个具有挑战性的目标。我们的第三个目标是研究由经典伪微分算子全纯族修饰的图的正则振幅的极点结构，并利用PROPs及其推广通过图的全称性质捕获它们的极点结构。

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.