Vortrag am

Donnerstag, 23.4.2015

Sylvie Paycha, Potsdam

" The residue of meromorphic functions with linear poles and the geometry of cones (based on joint work with Li Guo and Bin Zhang)"


Germs of meromorphic functions with linear poles at zero naturally arise in various contexts in mathematics such as when evaluating mutiple zeta functions at non negative integers or counting integer points on cones as well as in quantum field theory when computing Feynman diagrams. These issues have in common that they require evaluating meromorphic functions at poles in a consistent manner; in particular the evaluation should factorise on products of functions with independent variables. This multiplicativity property is reminiscent of locality in quantum field theory; two events far apart should be measured indpendently.
We provide a decomposition of the algebra of such germs into the holomorphic part and a linear complement by means of an inner product using the geometry of cones and associated fractions in an essential way.
This decomposition yields a renormalisation map given by the projection onto the holomorphic part, which enjoys the above mentioned multiplicative property.
Using this decomposition, we generalise the graded residue on germs of meromorphic functions in one variable to a graded residue on germs of meromorphic fractions in several variables with linear poles at zero and prove that it is independent of the chosen inner product. This residue applied to exponential discrete sums on cones yields the corresponding exponential integrals on cones.