On the curvature of discrete spaces

How can a discrete space be possibly curved? Because it shares some properties with Riemannian manifolds with particular curvature structures. In fact, in Riemannian geometry, sectional and Ricci curvature are central notions. They are defined infinitesimally, through first and second derivatives of the metric tensor. Curvature inequalities, like Sec 0, or Ric >0, have geometric consequences, for the angles in geodesic triangles, convexity properties of the distance function, the divergence or convergence of geodesics, the growth of the volume of geodesic balls, the eigenvalues of the Laplace-Beltrami operator, or coupling properties of Brownian motion. In fact, it turns out that some of those local properties are equivalent to certain curvature bounds. Since such properties are also meaningful on metric spaces more general than Riemannian manifolds, we can use them to define corresponding curvature inequalities on such more general spaces, and to explore their consequences and mutual relations.

In this talk, I shall introduce or discuss some such curvature inequalities on suitable classes of metric spaces, in particular discrete ones, and explore their geometric consequences.