# Schedule of the Workshop "Interactions between operads and motives"

## Wednesday, September 14

10:30 - 11:00 |
Registration & Welcome coffee |

11:00 - 12:00 |
Benoit Fresse: Rational homotopy theory, the little discs operads and graph complexes (Lecture 1) |

12:00 - 13:50 |
Lunch break |

13:50 - 14:50 |
Joseph Ayoub: Motivic Galois groups (Lecture 1) |

15:00 - 16:00 |
Johan Alm: Brown's dihedral moduli space and freedom of the gravity operad |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Markus Spitzweck: Mixed Tate motives and fundamental groups (Lecture 1) |

afterwards |
Reception |

## Thursday, September 15

09:30 - 10:30 |
Benoit Fresse: Rational homotopy theory, the little discs operads and graph complexes (Lecture 2) |

10:30 - 11:00 |
Group photo and coffee break |

11:00 - 12:00 |
Joseph Ayoub: Motivic Galois groups (Lecture 2) |

12:00 - 15:00 |
Lunch break and free time |

15:00 - 16:00 |
Claire Glanois: Periods of the motivic fundamental groupoid of ℙ^{1} ∖ {0, μ_{N}, ∞} |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Markus Spitzweck: Mixed Tate motives and fundamental groups (Lecture 2) |

## Friday, September 16

09:30 - 10:30 |
Benoit Fresse: Rational homotopy theory, the little discs operads and graph complexes (Lecture 3) |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Joseph Ayoub: Motivic Galois groups (Lecture 3) |

12:00 - 13:50 |
Lunch break |

13:50 - 14:50 |
Philip Hackney: Higher cyclic operads |

15:00 - 16:00 |
Markus Spitzweck: Mixed Tate motives and fundamental groups (Lecture 3) |

16:00 - 16:30 |
Tea and cake, end of workshop |

# Abstracts

## Johan Alm: Brown's dihedral moduli space and freedom of the gravity operad

Ezra Getzler's gravity cooperad is formed by the degree-shifted cohomology groups of the open moduli spaces M_{0,n}. Francis Brown introduced partial compactifications of these moduli spaces, denoted M_{0,n}^{δ}. We prove that the (nonsymmetric) gravity cooperad is cofreely cogenerated by the cohomology groups of Brown's partial compactifications, and discuss some implications of this result. The research is joint with Dan Petersen.

## Joseph Ayoub: Motivic Galois groups

## Benoit Fresse: Rational homotopy theory, the little discs operads and graph complexes

The little cubes operads (and the equivalent little discs operads) were introduced by Boardman-Vogt and May for the study of iterated loop spaces. The study of the little cubes operads has been completely renewed during the last decade and new applications of these objects have been discovered in various fields of algebra and topology. To cite one application, one can prove that the spaces of compactly supported embeddings of Euclidean spaces modulo immersions have a description in terms of mapping spaces associated to the little discs operads. This result represents the outcome of a series of works by Sinha, Arone-Turchin, Dwyer-Hess and Boavida-Weiss on the Goodwillie-Weiss calculus of functors.

The goal of my talk is to explain that the rational homotopy of mapping spaces associated to the little discs operads can be determined by graph complexes. This computation can also be performed for the spaces of homotopy automorphisms of the little discs operads. The homology of the graph complex associated to these homotopy automorphism spaces just reduces to the Grothendieck-Teichmüller group in the case of the 2-dimensional little discs. The proof of these results relies on a study of the rational homotopy of the little discs operads which I will also explain in my talk.

## Claire Glanois: Periods of the motivic fundamental groupoid of ℙ^{1} ∖ {0, μ_{N}, ∞}

Cyclotomic multiple zeta values (CMZV) are particularly interesting examples of periods (in the sens of Kontsevich-Zagier) and a fruitful recent approach is to look at their motivic version (MCMZV), which are motivic periods of the fundamental groupoid of ℙ^{1} ∖ {0, μ_{N}, ∞}. Notably, MCMZV have a Hopf comodule structure, dual of the action of the motivic Galois group on these specific motivic periods. After introducing motivic periods, and more particularly MCMZV, we will highlight how the explicit combinatorial formula of the coaction (given by Goncharov, Brown) enables, via the period map (isomorphism under Grothendieck’s period conjecture), to deduce results on CMZV. Then, we will apply some Galois descents ideas to the study of these motivic periods, and examine how periods of the fundamental groupoid of ℙ^{1} ∖ {0, μ_{N'}, ∞} are embedded into periods of π_{1}(ℙ^{1} ∖ {0, μ_{N}, ∞}), when N′ | N.

## Philip Hackney: Higher cyclic operads

## Markus Spitzweck: Mixed Tate motives and fundamental groups

In the first part we review representation theorems for triangulated categories of Tate motives in terms of modules over Adams graded E_{∞}-algebras. In the second part we show how these algebras can be used to define affine derived fundamental groups for Tate motives. In the third part we apply this theory to geometric fundamental groups. Throughout we motivate our constructions with topological examples.