# Málaga & Topology Meeting @ Universidad de Málaga

I will present several algebraic models for the real/rational homotopy types of (ordered) configuration spaces of points and framed points in a manifold. These models can be used to establish real/rational homotopy invariance of configuration spaces under dimensionality and connectivity assumptions. Moreover, the collection of all configuration spaces of a given manifold has the structure of a right module over some version of the little disks operad, and the algebraic models are compatible with this extra structure. The proofs all use ideas from the theory of operads, namely Kontsevich’s proof of the formality of the little disks operad and – for oriented surfaces – Tamarkin’s proof of the formality of the little 2-disks operad. (Based on joint works with Campos, Ducoulombier, Lambrechts, and Willwacher.)