Pure Maths Colloquium - Curvatures on graphs: Which graphs look like spheres? - Prof Norbert Peyerimhoff (Durham) Seminar

Event details

Abstract: Curvature is a fundamental mathematical concept being investigated and promoted in pioneering work by Gauss and Riemann. A natural and challenging problem is to find suitable curvature notions for other spaces like graphs. In this talk we will present one solution: Ollivier Ricci curvature for graphs and based on optimal transport. A fundamental result in the classical setting of Riemannian manifolds is that stricyly positive Ricci curvature implies boundedness of the space with an explicit upper diameter bound (Bonnet-Myers Theorem). Moreover, the diameter bound is assumed only for the round spheres (Cheng's Rigidity Theorem). Analogous results hold for Ollivier Ricci curvature on graphs and we will present a list of all graphs for which the curvature-diameter bound is assumed. In other words, we will classify all graphs which look like round spheres. This result is joint work with David Cushing, Supanat Kamtue, Jack Koolen, Shiping Liu and Florentin Muench.