Algebraic topology in the mesoscopic regime
Speaker: Antonio Rieser (CONACYT-CIMAT, A.C.)
Occasion: Topological Data Analysis
Date: April 27, 2021
Abstract: There have been a number of attempts to extend the realm of application of algebraic topological tools to discrete spaces such as graphs, digital images, and point clouds, which one more typically encounters in computer science and data analysis. In each of these theories, one of two strategies has typically been taken. In topological data analysis, one usually replaces the original space with one or more topological spaces that one hopes will retain the relevant topological information in the original set. In various approaches to discrete or digital topology, we find instead different attempts to develop algebraic topology from scratch for some class of discrete objects of interest, proceeding largely by analogy with classical algebraic topology. In this work, we propose a third option: we generalize algebraic topology to categories which contain both the topological spaces classically treated by classical homotopy theory, but which also include as objects the more discrete and combinatorial spaces of interest in applications. The advantage here is that there are now non-trivial ‘continuous maps’ from classical topological spaces to the discrete spaces (given the appropriate structure), and one may then compare the resulting topological invariants on each side functorially. We find that there are a number of possible such categories, each with its own particular homotopy theory and associated homologies, and, additionally, that there is a generalization of the coarse category which allows finite sets to be non-trivial (i.e. not ‘coarsely’ equivalent to a point). We will give an overview of these theories and several applications, discussing the advantages and disadvantages of each.