Abstract: This talk is primarily expository and aimed at graduate students and newcomers to homotopical algebra. We will introduce the language of bidifferential complexes--graded vector spaces equipped with both a differential and a codifferential--and use it to discuss deformation retractions and an algebraic form of the Hodge decomposition.

We will then examine in more detail the classical setting of differential forms on a Riemannian manifold, focusing on the homotopy transfer of the wedge product, which yields a complete invariant of its rational homotopy type.

In a subsequent talk, I plan to move beyond the Riemannian case and consider the Lorentzian setting, where physical information is also encoded in the transferred structure.