Some recent results on Ricci curvature on graphs

Ricci curvature plays a very important role in the study of Riemannian manifolds. In recent years, various adaptations of Ricci curvature emerged on graphs such as Bakry-Emery, Ollivier Ricci, Lin-Lu-Yau modified Ollivier Ricci and, more recently, Erbar-Maas. In my research, it turned out to be useful to develop an interactive applet to study some of these various notions. The main emphasis of this talk will be on Ollivier Ricci curvature, a notion based on optimal transport, and a modification proposed by S.T. Yau and co-authors. We will discuss various properties of this curvature notion like, e.g., dependence on the idleness parameter, behaviour under Cartesian products, and results for special families of graphs like Cayley graphs of Coxeter groups and strongly regular graphs, and graphs with strictly positive curvature. If time permits, we will indicate similarities with some of the other discrete curvature notions like Bakry-Emery curvature.