Cantor combinatorics and almost finiteness

TL;DR

This survey introduces a continuous analogue of Borel combinatorics focusing on almost finiteness, and explores its applications to spectral convergence of graph Laplacians, emphasizing an algorithmic perspective.

Contribution

It presents a continuous version of Borel combinatorics centered on almost finiteness and demonstrates its utility in spectral graph theory.

Findings

Almost finiteness provides a new framework for continuous Borel combinatorics.

The theory aids in understanding spectral convergence of graph Laplacians.

Abstract

In this survey we give a concise introduction to a continuous version of Borel combinatorics. Our approach will have a certain algorithm-theoretic nature and we will give special emphasis to the notion of almost finiteness introduced by Matui as a continuous analogue of Borel hyperfiniteness. We also show how the theory can be used to study spectral convergence for graph Laplacians.