Operads and Moduli of Curves

I shall describe how the Gravity and the Hypercommutative Operads, related to the cohomology of the Moduli of genus 0 curves, are Koszul dual to each other. Then we shall see how this duality can be exploited to determine the symmetric group equivariant cohomology of the compactified moduli space of genus 0 curves. We shall use similar techniques to do some cohomology computations that will prove the sharpness of an upper-bound in a conjecture by Eduard Looijenga.