- This meeting will take place in Paris (both at Université Sorbonne Paris Nord and at the École Normale Supérieure de Paris) from October 8th (3pm) to October 10th (1pm)

- Joana Cirici (Universitat de Barcelona)
- Coline Emprin (Université Sorbonne Paris Nord and École Normale Supérieure)
- Geoffroy Horel (Université Sorbonne Paris Nord)

- Pedro Boavida de Brito (Instituto Superior Técnico, Universidade de Lisboa)
- Marie-Camille Delarue (Université Paris Cité)
- Connor Malin (MPIM Bonn)
- Samuel Muñoz (University of Cambridge)
- Hugo Pourcelot ( Università degli studi di Firenze)
- Jakob Ulmer (Université Sorbonne Paris Nord)
- Bruno Vallette (Université Sorbonne Paris Nord)
- Adela Zhang (University of Copenhagen)

8 October |
15:00-16:00 | TALK 1 | |

16:00-16:30 | Coffee break | ||

16:30-17:30 | TALK 2 | ||

9 October |
10:00-11:00 | TALK 3 | |

11:00-11:30 | Coffee break | ||

11:30-12:30 | TALK 4 | ||

Lunch | |||

14:30-15:30 | TALK 5 | ||

11:00-11:30 | Coffee break | ||

16:00-17:00 | TALK 6 | ||

10 October |
10:00 - 11:00 | TALK 7 | |

11:00 - 11:30 | Coffee break | ||

11:30 - 12:30 | TALK 8 |

The location of the talks will be made precise soon.

Titles and abstracts: TBA.

The interaction of the little disks operad E_n and Koszul duality has been, and continues to be, studied throughout the past 30 years. In particular, what is the relation between $E_n$ and its Koszul dual? We survey the history of this problem, and then describe our recent construction of an explicit, simple, and geometrically defined map which witness the equivalence of the $E_n$ operad and its Koszul dual, in the category of spectra.

For given p less than d, the space of "long knots" is the embedding space rel boundary of the p-disc into the d-disc. By work of Boavida de Brito—Weiss et al., when d-p is at least 3, its homotopy type is closely related to that of the space of operad maps from E_p to E_d, where E_n denotes the little n-discs operad. In this talk, I will explain a computation of this homotopy type away from the prime 2 and in the so-called "concordance embedding stable range" which, by developments of Goodwillie—Krannich—Kupers, is at least 2d-p-5. The description features objects internal to surgery theory as well as relative algebraic K-theory/topological cyclic homology, appearing as part of an analogue for embeddings of a theorem of Weiss—Williams. I will also explain how this computation recovers rationally the 0- and 1-loop order parts of the hairy graph complex of Fresse—Turchin—Willwacher.

The bar spectral sequence for algebras over a spectral operad relates Koszul duality phenomena in several contexts. We apply this spectral sequence to the Koszul dual pair given by the (non-unital) E_∞ operad and the spectral Lie operad. When the input are trivial E_∞-HF_p algebras, we obtain the structure of operations on mod p TAQ cohomology and the homotopy groups of spectral partition Lie algebras, building on the work of Brantner–Mathew. In the colimit, the unary operations are Koszul dual to the Dyer–Lashof algebra in the sense of Priddy. There is also a shifted restricted Lie structure that can be detected by the homotopy fixed points spectral sequence, reflecting the Koszul duality between the commutative operad and the shifted Lie operad over F_p