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.