Rational homotopy type of the moduli space of stable rational curves

In 2004, Manin asked whether the cohomology of the moduli space of stable rational curves with n marked points (= the Deligne-Mumford compactification of the moduli space of smooth genus zero curves with n marked points) is a Koszul algebra. This question remained open since. I shall present a solution to it, proving that the answer is positive for all n. An immediate consequence of my result is an explicit description of the rational homotopy Lie algebras of these spaces by generators and relations. Time permitting, I shall discuss some generalizations and modifications of this result.