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Combinatorial Dyson-Schwinger equations and polynomial functors

Joachim Kock
(Universitat Autònoma de Barcelona)
Seminar
Monday, 8 May, 2017, at 13:15-14:00, in Koll. G (1532-214)
Abstract:

I will explain what are the combinatorial Dyson-Schwinger equations, first by briefly outlining their role in quantum field theory, and then by showing that they can be understood in terms of some elementary category theory. The formulation of the equations by Bergbauer and Kreimer starts with a Hopf algebra (of trees or graphs), and a collection of Hochschild 1-cocycles, and one of their main theorems is that the solution spans a sub Hopf algebra isomorphic to the Faa di Bruno Hopf algebra (the Hopf algebra dual to composition of formal power series). The new categorical interpretation starts very abstractly with a polynomial fixpoint equation of sets, nothing more. I will explain how this data canonically generates trees, Hopf algebras, Hochschild 1-cocycles, and Faa di Bruno formula. This reveals close connections with inductive data types in program semantics, which I can explain if time permits.

Note: This seminar is aimed at a general audience of mathematicians 

Organised by: QGM
Contact person: Jørgen Ellegaard Andersen