Institute of Mathematics University of Potsdam

Starting from the principle of locality in quantum field theory, which
states that an object is influenced directly only by its immediate

surroundings, I will first briefly review some features of the notion of
locality arising in physics and mathematics. These are then encoded
in  locality relations, given by symmetric binary relations whose graph
consists of pairs of "mutually independent elements".

I will mention challenging questions that arise from  enhancing algebraic
structures to their locality counterparts, such as i) when  is the quotient
of a locality vector space by a linear subspace, a locality vector space, if
equipped with the quotient locality relation,  ii) when does  the locality
tensor product of two locality vector spaces  define a locality vector
space. These are discussed in recent joint work  with Pierre Clavier, Loïc
Foissy and Diego López.

Locality morphisms, namely maps that factorise on   products of  pairs of
"mutually independent" elements, play a key role in the context of
renormalisation in
multiple variables. They include "locality evaluators", which we use to

consistently evaluate meromorphic germs in several variables at
their poles. I will  also report on recent joint work with Li Guo and Bin
Zhang. which gives a classification of locality evaluators on certain
classes of algebras of meromorphic germs.

How to evaluate  meromorphic germs at their poles while preserving a
locality principle reminiscent of locality in QFT is a    question that
lies at the heart of  pQFT. It further  arises in other disguises in
number theory, the combinatorics on cones and toric geometry. We
introduce an abstract notion of locality and a related notion of
mutually independent meromorphic germs in several variables. Much in the
spirit of Speer's generalised evaluators in the framework of analytic
renormalisation, the question then amounts to extending the ordinary
evaluation at a point  to  certain algebras of meromorphic germs, in
such a way that the extension  factorises  on mutually independent
germs. In the talk, we shall describe a family of such extended
evaluators  and show that modulo a Galois type  transformation, they
amount to a minimal subtraction scheme in several variables.
This talk is based on ongoing joint work with Li Guo and Bin Zhang.