Diameter bounds in graph curvature, David Cushing
A fundamental question in geometry is in which way local properties determine the global structure of a space. One famous result of this kind is the celebrated Bonnet-Myers Theorem. I will survey generalisations of this theorem to graphs and the classification problem of when this result is "rigid". I will also discuss some new advancements in this area and pose some open problems.
Bakry-Émery curvature and its connection to Entropic Ricci curvature, Supanat Kamtue
Motivated from Bochner's formula in Riemannian geometry, Bakry-Émery curvature is a notion of Ricci-type curvature introduced in the setting of discrete spaces by Elworthy (1989), Schmuckenschlager (1996) and Lin-Yau (2010). Our work proposes the reformulation of the Bakry-Emery curvature as the smallest eigenvalue of a symmetric matrix, namely the "curvature matrix". As an application, we derive a simple formula for the curvature of Cartesian products.
There is another notion of discrete RIcci curvature, namely entropic Ricci curvature, which is introduced by Erbar and Maas (2012) in spirit of the famous work by Lott-Sturm-Villani that the entropy functional is convex along Wasserstein geodesics. In this talk, I will point out some connection between Bakry-Émery curvature and this entropic Ricci curvature via the Bochner's formula.
Olliviers' Ricci curvature on hypergraphs, Riikka Kangaslampi
In this talk I will define Ollivier's Ricci curvature on hypergraphs, giving two different formulations for the definition, and discuss a few preliminary results. I will also present a literature review of the current knowledge on the subject.
Computing the Ricci-Flatness of graphs and Steinerberger curvature, Erin Law
We discuss two curvature notions recently added to the graph curvature calculator and compare them to Ollivier-Ricci curvature. The notion of Ricci flat graphs was introduced in 1996 by Chung and Yau in connection to a logarithmic Harnack inequality . We will show that self centred Ollivier-Ricci Bonnet-Myers graphs are Ricci-Flat. The recently introduced Steinerberger curvature is based on equilibrium measures, we compute it for a large class of families and compare our findings to Ollivier-Ricci curvature.
Bakry-Émery curvature matrices for discrete magnetic Laplacians, Shiping Liu
I will explain how to reformulate the Bakry-Emery curvature for graph connection Laplacians as the smallest eigenvalue of a curvature matrix. Indeed, at each vertex, we have a family of unitary equivalent curvature matrices. Hopefully, this provides further insights to the previous works on Bakry-Emery curvature for graph Laplacians by Siconolfi, and by Cushing, Kamtue, Peyerimhoff and myself. This talk is based on a joint work with Chunyang Hu (USTC).
Mixing time and expansion of non-negatively curved Markov chains, Florentin Münch
TBA
A curvature flow for weighted graphs based on the Bakry-Emery calculus, Norbert Peyerimhoff
The aim of this talk is to introduce a time continuous curvature flow for Markovian weighted graphs and to discuss some of its properties. A weighted graph is Markovian if the weights on the directed edges represent transition probabilities of a (potentially lazy) random walk. We allow vanishing transition probabilities along edges, in which case we call the weighting scheme degenerate. The Bakry-Emery calculus, motivated by Bochner's formula for Riemannian manifolds, can be used to define a natural Ricci curvature notion on the vertices of such a weighted graph. The problem of calculating this vertex curvature can be reformulated, in the case of a nondegenerate weighted graph, as the problem to calculate the smallest eigenvalue of a specific symmetric matrix (this fact was discovered independently by Viola Siconolfi and in earlier work with D. Cushing, S. Kamtue and Sh. Liu). We use this matrix to introduce an associated curvature flow. By normalising this curvature flow such that it preserves the Markovian property, it turns out that the limits of this curvature flow (as time tends to infinity) are always curvature-sharp weighted graphs. This is joint work with D. Cushing, S. Kamtue, Sh. Liu, F. Muench, and B. Snodgrass.
Ricci curvature of graphs from Coxeter theory, Viola Siconolfi
I will talk about a notion of curvature for graphs introduced by Schmuckenschläger which is defined as an analogue of Ricci curvature. I will present some general results on the computation of the discrete Ricci curvature for locally finite graphs. I will then focus on some graphs associated with Coxeter groups: Bruhat graphs, weak order graphs and Hasse diagrams of the Bruhat order.