Configuration spaces of manifolds, that is, ordered finite collections of pairwise distinct points, are classical yet intriguing objects in algebraic topology. They admit a rich algebraic structure coming from the theory of operads. In this mini-course, I will explain how this structure is defined, and how one can show, using the extra algebraic structure, that the real homotopy type of the ambient manifold completely determines the real homotopy type of the configuration spaces under some hypotheses.
These talks are based on joint works with Campos, Ducoulombier, Lambrechts, and Willwacher.