David Aristoff's homepage Title: Discrete curvature and the shape of networks

Abstract: A fundamental computational challenge in understanding the geometry of networks is that global computations become prohibitively expense on large networks. One desires, then, a way of certifying certain geometric properties of graphs by looking at only local information (perhaps involving only a bounded number of vertex neighborhoods.) Inspiration for such can come from Riemannian geometry. One of the most important local properties of a manifold is its curvature. Lower bounds on the curvature of a Riemannian manifold yield a number of strong results about the manifold's properties. Defining an appropriate analogue in the case of graphs, however, has proven a difficult challenge. None the less, a number of recent results have made progress in this direction. Work of Bauer, Lippner, Lin, Mangoubi, Yau and the speaker introduced a notion of curvature which enabled them prove an analogue of the celebrated Li-Yau inequality and derive from it a number of geometric consequences. Further work of Lin, Liu, Yau and the speaker has gone further, enabling the derivation of volume doubling and Poincare inequalities, along with other consequences, from non-negative curvature. We will talk about some of these results, along with some other recent related advances.