La onzième session des Après-midi de Topologie, co-organisées par Christian Ausoni (LAGA), Geoffroy Horel (LAGA), Najib Idrissi (IMJ-PRG) et Muriel Livernet (IMJ-PRG).
Denis Lyskov : Graphical generalisation of operads and configuration spaces
Operads and related structures play a significant role in algebraic topology and homotopy theory, particularly in the study of configuration spaces of manifolds. In this report, we will discuss generalised configuration spaces indexed by graphs in which some marked points may coincide and contractads - new operad-like structures that underlie these configuration spaces.
Operations in contractads are indexed by connected graphs and composition rules are numbered by contractions of connected subgraphs. We will show that many classical operads, such as the operad of commutative/associative/Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves admit generalisations to contractads. In most of the report, we will discuss which results on little disks operads are valid in the contractad case, as well as open problems.
Nina Otter : What is the dimension of a graph? A view from topological data analysis
Dimension theory of graphs is a field that has been developed over the last 30 years and tries to translate notions of dimension from the continuous to the discrete setting. When applied to graphs with weights assigned to their edges, these notions of dimension are not always well-behaved. In this talk I will give an overview of the state-of-the-art, explain what can go wrong, and how methods from Topological Data Analysis may come to our rescue.
This talk is based on work in progress, as well as the article https://arxiv.org/abs/2506.15236
Nikola Tomic : AKSZ Theorem for shifted Poisson structures
In algebraic geometry, it is classical to study moduli stacks of some objects or structures. In general, those stacks are highly singular so it is really hard to study them. However, during the last two decades, derived geometry was created and developed. This theory allows one to bypass singularity problems and to study moduli stacks in a flexible way.
The AKSZ theorem (named after Alexandrov–Kontsevich–Schwarz–Zaboronsky) in shifted symplectic geometry is an important theorem of derived geometry establishing existence of shifted symplectic structures on many moduli stacks given a symplectic structure on some easier stacks, allowing one to properly study invariant on them (using the symplectic structure), and quantize them. In order to quantize a shifted symplectic moduli stack, one have to relate it to Poisson structures. In fact, a shifted symplectic structure is a particular case of a shifted Poisson structure and then one use Kontsevitch formality to produce a quantization. It is then natural to ask whether or not, on some moduli stacks, there exists shifted Poisson structures coming from easier shifted Poisson stacks. In this talk, I will expose the theory of shifted Poisson geometry and sketch a proof of the AKSZ theorem for shifted Poisson structures. | 
