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

Marzieh Eidi, Juergen Jost, Dong Zhang

TL;DR

The paper surveys deep connections between discrete and continuous geometric theories, focusing on Hodge-Eckmann structures, Morse-Witten-Floer viewpoints, and Cheeger-type inequalities. It highlights how simplicial complexes, graphs, and manifolds share parallel Laplacians, spectral properties, and variational characterizations, and it analyzes convergence of discrete models to smooth manifolds. Key contributions include formalizing higher-order Cheeger relations via signed graphs constructed from simplicial complexes, and clarifying how discrete Morse theory and its Witten-type deformations parallel the smooth theory. The discussion also identifies open problems in stochastic processes on higher-dimensional chains, convergence rates, and the extension of Morse-Floer-type constructs to broader dynamical settings, with implications for topology, geometry, and data science.

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.

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

TL;DR

The paper surveys deep connections between discrete and continuous geometric theories, focusing on Hodge-Eckmann structures, Morse-Witten-Floer viewpoints, and Cheeger-type inequalities. It highlights how simplicial complexes, graphs, and manifolds share parallel Laplacians, spectral properties, and variational characterizations, and it analyzes convergence of discrete models to smooth manifolds. Key contributions include formalizing higher-order Cheeger relations via signed graphs constructed from simplicial complexes, and clarifying how discrete Morse theory and its Witten-type deformations parallel the smooth theory. The discussion also identifies open problems in stochastic processes on higher-dimensional chains, convergence rates, and the extension of Morse-Floer-type constructs to broader dynamical settings, with implications for topology, geometry, and data science.

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.

Paper Structure

This paper contains 28 sections, 9 theorems, 112 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Structures and questions
  3. Simplicial complexes
  4. The topology of simplicial complexes
  5. The Eckmann Laplacian
  6. Riemannian manifolds
  7. Rayleigh quotients
  8. $p$-Laplacians
  9. Lovász extension
  10. Analogies and convergence
  11. Morse theory and flows
  12. Morse theory on Riemannian manifolds
  13. The Witten Laplacian
  14. Morse-Smale systems and the Floer boundary operator
  15. Combinatorial Morse theory
  16. ...and 13 more sections

Key Result

Theorem 2.1

Suppose $d=0$. For any $k$, the $k$-th minimax eigenvalue $\lambda_k(L^{up}_{0,p})$ is locally Lipschitz continuous with respect to $p$, and moreover,

Figures (7)

  • Figure 3.1: A Morse function on $S^2$
  • Figure 3.2: An example that does not satisfy the condition for the boundary operator
  • Figure 3.3: Resolving the problem of Fig.\ref{['fig1a']} by adding a pair of critical points, a maximum and a saddle
  • Figure 3.4: A Morse-Smale dynamical system on $S^2$ with a periodic orbit $O$
  • Figure 3.5: Replacing the periodic orbit in Fig. \ref{['fig2']} by two heteroclinic orbits, resulting in the situation of Fig. \ref{['fig1']}
  • ...and 2 more figures

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2: Lovász
  • Theorem 2.3: Lovász
  • Theorem 2.4
  • Remark 2.1
  • Remark 3.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1
  • Remark 3.2
  • ...and 7 more