High-Dimensional Expansion (HDX)

High-dimensional expansion (HDX) is a generalization of expansion in graphs to higher dimensions (i.e., hypergraphs). In recent years, HDXs have already seen many impressive applications in areas such as error-correcting codes, bounding the mixing time of Markov chains and probabilistically checkable proofs (PCPs), and explicit constructions of vertex expanders. In this first of three lectures from our ICM satellite conference, Mitali Bafna defines the spectral notion of HDX, proves the trickle-down theorem, and defines random walks on HDX. See also Part 2 and Part 3, in which she shows the Kaufman–Oppenheim construction of HDX.