Interactions between complex, real, and tropical geometries: an example

Denote by ${\mathcal{M}}_{g,n}$ the moduli space parametrizing a genus $g$ smooth complex curves with $n$-marked points. These moduli spaces are one of the central objects in modern science which appears in algebraic geometry, arithmetics, quantum physics… The challenging problem which we will discuss is the computation of the cohomology of ${\mathcal{M}}_{g,n}$

In my talk we will attack this problem from point of view of a triad: $$({\mathcal{M}}_{2g+n-1}^{\mathbb{R}},{\mathcal{M}}_{g,n},M_{g,n}^{trop})$$ Here ${\mathcal{M}}_{2g+n-1}^{\mathbb{R}}$ is a moduli space of real curves and $M_{g,n}^{trop}$ is a moduli space of tropical curves. I will explain the correspondences between different moduli spaces of the triad. In particular we relate the cohomology of ${\mathcal{M}}_{g,n}$ to the cohomology of combinatorial objects called Kontsevich-Penner ribbon graph complex and the hairy Kontsevich graph complex. Further, we will discuss the correspondence between different structures on the elements of the triad. The talk is an overview of the works of K. Costello '07, S. Merkulov and T. Willwacher '15, M. Chan S. Galatius and S. Payne '18-'19, A. Andersson T. Willwacher and M. Zivkovič '20, T. Willwacher and S. Payne '21, A.K. '20-'22.