"Polynomial functors over groupoids, and combinatorial
Dyson-Schwinger equations"
Polynomial functors are essentially functors defined in terms of
sums, products, and exponentiation. They can be seen as a
categorification of elementary arithmetic, but they also
constitute a general machinery for encoding and handling
combinatorial structures and data types. Groupoid coefficients
rather than set coefficients are needed to transparently handle
symmetries, like those occurring in Feynman graphs.
The real Dyson-Schwinger equation are the 'quantum equations of
motion'. I will only talk about their combinatorial skeleton,
Kreimer's combinatorial Dyson-Schwinger equations, which are
fixpoint equations whose solutions are certain series of graphs
or trees.
I will explain how any polynomial endofunctor P generates a free
monad, whose operations form a groupoid of P-trees, which is a
solution to an abstract combinatorial Dyson-Schwinger equation
X = 1+P(X), and satisfies a Faˆ di Bruno formula in the
Connes-Kreimer bialgebra of P-trees. Analogous results for
Feynman graphs are obtained by specialising to certain
endofunctors P given in terms of interaction labels and 1PI
primitive graphs. Many of the ideas involved originate in the
theory of inductive data types. If time permits, I may say
something about that.