Topology, graphs, and data

This talk will be an introduction to the emerging area of discrete homotopy theory, which applies intuitions and techniques from the continuous setting to discrete objects such as graphs. It has found a broad range of applications, both within and outside mathematics, including to matroid theory, hyperplane arrangements, and data analysis.

I will discuss two of my own contributions to discrete homotopy theory, one more theoretical and one more applied. The first is a proof, joint with D. Carranza (Compos. Math., 2024), of the conjecture by E. Babson, H. Barcelo, M. de Longueville, and R. Laubenbacher that discrete homotopy groups can be topologically realized. The second, joint with N.
Kershaw (arXiv:2506.15020), builds on this result and introduces a new method of data analysis, which we call persistent discrete homology. We
show that in addition to its utility for clustering, it can detect other geometric features of a data set. It is furthermore highly noise resistant, and as such provides a powerful alternative to the usual methods of (unsupervised) machine learning, especially in areas subject to high uncertainty, such as seismology or crime linkage.