Moduli Spaces of Nodal Curves from Homotopical Algebra

I will discuss how the Deligne-Mumford compactification of curves arises from the uncompactified moduli spaces of curves as a result of some algebraic operations related to (pr)operadic structures on the moduli spaces. I will describe how a variation of this naturally gives rise to another new partial compactification of moduli spaces curves. Time permitting, I will indicate how it is related to secondary operations on symplectic cohomology and discuss some ongoing work in this direction.