Cubical setting for discrete homotopy theory

Discrete homotopy theory, introduced around 20 years ago by H. Barcelo and collaborators building on the work of R. Atkin from the mid-seventies, is a homotopy theory of (simple) graphs. As such, it applies techniques previously employed in the "continuous" context to study discrete objects. It has found applications both within and outside mathematics, including: matroid theory, hyperplane arrangements, topological data analysis, time series analysis, and quite concretely in understanding how social interactions between preschoolers impact their academic performance.
Recently, discrete homotopy theory has seen remarkable progress leading to proofs of several longstanding conjectures.
This talk will be an introduction to discrete homotopy theory and will report on a proof, joint with D. Carranza (arXiv:2202.03516), of the conjecture of E. Babson, H. Barcelo, M. de Longueville, and R. Laubenbacher from 2006.