Research

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

1. 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)
2. 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)
3. 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)
4. 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)
5. D. Cushing and S. Kamtue, Long-scale Ollivier Ricci curvature of graphs, Anal. Geom. Metr. Spaces 7 (2019), no. 1, 22–44. (DOI)