Configurations spaces, algebraic topology and operads (2022−2023)

Universidad Nacional de la Patagonia San Juan Bosco (Puerto Madryn, Argentina)
2023 CIMPA Research School “Crossroads of geometry, representation theory and higher structures”

I plan to explain how two classical objects in algebraic topology are related: configuration spaces, and operads. Configuration spaces, on the one hand, consist in collections of pairwise distinct points in a given manifold. The study of these spaces is a classical problem in algebraic topology. Operads, on the other hand, are algebraic objects whose representations are categories of algebras (e.g., associative algebras, commutative algebras, or Lie algebras). The intimate relationship between these two objects comes in the form of the little disks operads, a certain family of operads which is central in many applications and which involves the configuration spaces of the Euclidean spaces. I will show how ideas that come from the theory of operads – namely, the proof of the formality of the little disks operads – is useful in order to settle the question of the rational homotopy invariance of configuration spaces.