Renormalisation of multivariate meromorphic germs; a geometric approach

Germs of multivariate meromorphic functions with linear poles at zero>  naturally arise  in various contexts,
in number theory (multizeta functions),  the combinatorics on cones (exponential sums) and also in quantum
field theory (Feynman diagrams). We want to  extend to multivariate germs, concepts and results known for
meromorphic germs in one variable, such as Laurent expansions and the residue. Going from one to several
variables introduces new difficulties, which we tackle using the geometry of cones in an essential manner.
We discuss> various applications of these results,   revisiting  Berline and Vergne's Euler-Maclaurin formula
on cones and   generalising the Jeffrey-Kirwan residue.

 This is based on joint work with  Li Guo and Bin Zhang.