From local discrete curvature notions of graphs to global properties of the underlying spaces

Norbert Peyerimhoff (Durham University) https://simons.berkeley.edu/talks/norbert-peyerimhoff-durham-university-2026-09-21 Local to Global Principles Two widely used Ricci curvature notions for general graphs are Ollivier and Bakry-Emery curvature. They are defined locally on the edges and vertices of a graph and they can be computed with very reasonable effort. In this talk, I will introduce these discrete curvature notions and discuss implications of curvature restrictions to global invariants like diameter and Laplace spectrum of the underlying graphs. These relations are motivated by similar results in the setting of Riemannian manifolds (Bonnet-Myers and Lichnerowicz), and it is natural to ask whether all spaces, for which these curvature/diameter and curvature/eigenvalue inequalities hold with equality, can be classified. If time permits, I will also mention such kind of results.