Expander graphs are sparse yet highly connected, with several closely related notions of expansion. High dimensional expanders (in short HDX) generalize these ideas from graphs to hypergraphs, where edges can contain more than two vertices.
This talk will introduce high-dimensional expansion, focusing on how spectral expansion generalizes beyond graphs. We will begin with a brief review of expansion in graphs and some of its properties. We will then move from hypergraphs and form simplicial complexes from these hypergraphs, which provide the setting for studying high-dimensional expanders. In this setting, we will define local spectral expanders, explain the intuition behind their definition, and explore how spectral expansion in local neighborhoods helps us understand the behavior of the larger structure.
High-dimensional expanders have applications in constructing locally testable codes and PCPs, counting combinatorial objects, and designing rapidly mixing random walks on hypergraphs.
