Multiple sums and integrals associated with planar Schröder trees and set compositions

11.11.2022, 11:00  –  Campus Golm, Haus 9, Raum 2.22 + Zoom
Arbeitsgruppenseminar Analysis

Yannic Vargas (Universität Graz)

We investigate and compare two different ways of associating multiple sums and multiple integrals to a certain class of planar rooted trees (Schröder trees). These give rise to characters on the algebra of planar
Schröder trees equipped with a shuffle product. They generalise the multiple integrals associated with binary trees considered by Patras and Ebrahimi-Fard and  specialise to characters on the Loday-Ronco algebra of binary trees.
A first construction uses the universal property of rooted trees in the spirit of previous work  of Clavier, Guo, Paycha and Zhang in the divergent case and Clavier's study  in the convergent setup.  An alternative approach uses the inclusion-exclusion principle  and the multiplicative property of sums and integrals over independent sets of variables.  Specifying the class of  summands of multiple sums associated to trees  leads to branched zeta functions, which generalise multiple zeta functions. We show that they  coincide with the planar version of arborified (or branched) zeta functions studied by Clavier in the convergent case. Criteria for convergence follow from a "flatening procedure" using a well-known connection between  the algebras of set compositions and of Schröder trees. This is joint work with Sylvie Paycha and Diego Lopez.

You are welcome to invite your friends and colleagues to join us! If you wish to attend the talks by Zoom, please contact Sylvie Paycha paycha@math.uni-potsdam.de for the login details.