Discrete homotopy theory is a branch of homotopy theory dealing with graphs. But aren't graphs homotopically just collections of circles? Yes, but we are looking at graphs through a different lens in discrete homotopy theory, which detects finer combinatorial data. Through this lens, the homotopy theory of graphs is much richer than the homotopy theory of circles.

This talk gives a whirlwind tour of this world, starting from the philosophy behind abstract homotopy theory, basic invariants ("discrete homotopy/homology groups"), what they are used for, and interesting phenomena. I will end with a recent contribution by me and my coauthors to the Discrete Homotopy Hypothesis (DHH), a major conjecture in this area.