I study various approaches to define Ricci curvature on discrete spaces (graphs and networks). The first notion is Ollivier Ricci curvature defined via the L1-transportation of two small balls. The second notion is Bakry-Émery curvature defined via Bochner’s formula and carré du champ operators. The third notion is the entropic Ricci curvature defined by Erbar-Maas via the displacement convexity of the entropyin in spirit of Lott-Sturm-Villani’s definition of Ricci curvature on metric measure spaces.
Ph.D. Thesis
Discrete curvatures motivated from Riemannian geometry and optimal transport: Bonnet-Myers-type diameter bounds and rigidity (archive) - under the supervision of Prof. Norbert Peyerimhoff, Durham University.
Publications
- D. Cushing, S. Kamtue, S. Liu and N. Peyerimhoff, Bakry-Émery curvature on graphs as an eigenvalue problem, Calc. Var. Partial Differential Equations 61 (2022), no. 2, Paper No. 62, 33 pp. (DOI)
- D. Cushing, S. Kamtue, R. Kangaslampi, S. Liu and N. Peyerimhoff, Curvatures, graph products and Ricci flatness, J. Graph Theory 96 (2021), no. 4, 522–553. (DOI)
- D. Cushing, S. Kamtue, J. Koolen, S. Liu, F. Münch and N. Peyerimhoff, Rigidity for the Bonnet-Myers for graphs with respect to Ollivier Ricci curvature, Adv. Math. 369 (2020), 107188, 53 pp. (DOI)
- D. Cushing, S. Kamtue, N. Peyerimhoff and L. Watson May, Quartic graphs which are Bakry-Émery curvature sharp, Discrete Math. 343 (2020), no. 3, 111767, 15 pp. (DOI)
- D. Cushing and S. Kamtue, Long-scale Ollivier Ricci curvature of graphs, Anal. Geom. Metr. Spaces 7 (2019), no. 1, 22–44. (DOI)