High dimensional expansion generalizes edge and spectral expansion in graphs to higher dimensional hypergraphs or simplicial complexes. Unlike for graphs, it is exceptionally rare for a high dimensional complex to be both sparse and expanding. The only known such expanders are number-theoretic or group-theoretic.
Their key feature is a local-to-global geometry, that allows deducing global information about the entire complex from local information in the neighborhoods / links. We will discuss some results known about these objects, and how their local-to-global geometry, shared also by PCPs, can potentially lead to new codes and proofs.