From the discrete to the continuous, from simplicial complexes to Riemannian manifolds. Approximating flows and cuts on manifolds by discrete versions

TL;DR

This survey explores how discrete structures like graphs and simplicial complexes approximate continuous Riemannian manifolds, focusing on theories like Hodge, Morse, and spectral analysis, and recent advances in their interrelations.

Contribution

It reviews current research on the connections between discrete and continuous geometric structures, highlighting recent developments and open questions.

Findings

Cheeger inequalities for higher-dimensional simplicial complexes

Floer-type constructions in dynamical systems

Disorientability of simplicial complexes

Abstract

Many fundamental structures of Riemannian geometry have found discrete counterparts for graphs or combinatorial ones for simplicial complexes. These include those discussed in this survey, Hodge theory, Morse theory, the spectral theory of Laplace type operators and Cheeger inequalities, and their interconnections. This raises the question of the relation between them, abstractly as structural analogies and concretely what happens when a graph constructed from random sampling of a Riemannian manifold or a simplicial complex triangulating such a manifold converge to that manifold. We survey the current state of research, highlighting some recent developments like Cheeger type inequalities for the higher dimensional geometry of simplicial complexes, Floer type constructions in the presence of periodic or homoclinic orbits of dynamical systems or the disorientability of simplicial complexes.