Ricci Flow on Discrete Structures

• Applicants: Professor Dr. Matthias Keller; Dr. Florentin Münch
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 549422058

On Riemannian manifolds the Ricci flow is widely considered as the canonical heat flow of the Riemann metric. While it is a classical notion going back to Hamilton in the 1980’s, it showed its full glory and power in the work of Perelman on the Poincar´e conjecture in the early 2000’s. In recent years there were various attempts in pure and applied mathematics to define corresponding notions for graphs. This has been preceded by tremendous efforts to put forward various discrete Ricci curvature notions and to investigate their geometric and analytic meaning. The geometrization of discrete objects has brought forward significant progress in recent years bringing together researchers from various communities such as combinatorics, geometry, probability, analysis as well as data and computer science. Two central motivations for discrete curvature can be highlighted as follows. First a pure mathematics point of view is to explore Perelman’s proof from a discrete perspective. This promises to provide the tools to tackle the multitude of topological questions arising from Perelman’s result. Second from a applied mathematics perspective the notion of clustering is most crucial and was already studied empirically using discrete Ricci flow. Developing a structural and rigorously founded understanding of clustering with the help of discrete Ricci flow is desirable for many communities. While these are both long term goals which go beyond this proposal, we aim to lay the mathematical foundations to pursue them in the future. There are three guiding questions which permeate the proposal: (a) What one can learn from continuous structures for discrete structures? (b) What one can learn from graphs for simplicial and cell complexes? (c) What is the relationship between different discrete Ricci curvature and Ricci flow notions? Indeed, on Riemannian manifolds there is one classical notion of Ricci curvature with a wide range of characterizations of lower bounds. In contrast, in the discrete setting, there is a multitude of analogues and bounds. They turn out to be non-equivalent due to the ambiguity of discrete gradient notions and on a technical level due to the lack of a chain rule for the Laplacian. In this regard a deep understanding has been developed for the analysis and geometry of discrete spaces. Thus, there are several well established discrete Ricci curvature notions. It should not come as a surprise that each of these discrete Ricci curvature notions comes with their own versions of Ricci flow. Many of these notions have already been used in specific applications such as community detection, network alignment, graph neural network rewiring. However, the theoretical investigation of these flows is still in its infancy.

• DFG Programme: Research Grants