Mikael Vejdemo-Johansson: Gröbner bases for operads

Gröbner bases hold a significant role in introducing computational and
computer based techniques to the study of commutative, and later non-
commutative, rings. At the core of the Gröbner basis approach is a
systematic way to find a new basis for the ring in question that
captures not only the additive, but also the multiplicative structure
of the ring.

In a recent paper, Dotsenko and Khoroshkin draw up the relevant
choices to construct a Gröbner basis theory for symmetric (and non-
symmetric by extension) operads over the category of vector spaces.
This allows for several interesting new techniques based on these
definitions. For starters, being a multiplicative basis, a quadratic
Gröbner basis forms a Poincare-Birkhoff-Witt basis, and thus, by a
result by Hoffbeck, in itself a proof of Koszularity for the operad in
question. Furthermore, having a Gröbner basis means, by standard
constructions, that there is a normal form for any expression, which
opens up the theory of operads for computational approaches and tools.

In this talk, we will go through the essential definitions and
intuitions leading up to the Dotsenko-Khoroshkin paper, and culminate
in an overview of a recent computer implementation of the methods of
Dotsenko-Khoroshkin, written by Dotsenko and the speaker.