You need to know: Graph, connected graph, infinite graph, degree of a vertex of a graph, bounded degree graph (for which there is a constant such that every vertex has degree at most C), path connecting 2 vertices, length of a path, notation for the length of the shortest path connecting vertices and in graph G.
Background: Let G be a connected graph with infinite vertex set V. For each pair we define a conductance such that and also unless x and y are connected by edge. The graph G together with the conductances is denoted and called a weighted graph. For , let . For , let . A function is called harmonic on A if for all . For and , let be the set of with . We say that the elliptic Harnack inequality holds for if there exists such that whenever , , and h is non-negative and harmonic in , then for all . We say that weighed graphs and with vertex sets and are roughly isometric if there is a function and constants and such that (i) for every there exists , such that , (ii) for all , and (iii) for all and .
The Theorem: On 5th October 2016, Martin Barlow and Mathav Murugan submitted to arxiv a paper in which they proved the following result. Let and be two connected bounded degree weighed graphs that are roughly isometric. Then the elliptic Harnack inequality holds for if and only if it holds for .
Short context: The elliptic Harnack inequality was proved by Moser in 1961 for some partial differential equations, but since that turned out to be useful in many other applications, for example for weighted graphs. The Theorem proves that this inequality is stable under rough isometries, resolving a long-standing open question. While we state the Theorem only for weighed graphs here, Barlow and Murugan actually proved it in much more general setting.
Links: Free arxiv version of the original paper is here, journal version is here.
Theorems of the 21st century 