Abstract:

High-dimensional expanders (HDX) arise from generalizing the notion of expansion for graphs to (higher-dimensional) simplicial complexes. Like their one-dimensional versions, HDX have proven to be useful in theoretical computer science, but only a few constructions are known at this point. In this talk, I will give an introduction to the world of high-dimensional expansion and explain the construction of the KMS complexes which are new examples of HDX. The construction uses the rich structure of Kac-Moody-Steinberg groups which are certain amalgamated products of unipotent subgroups of Chevalley groups. Locally, the complexes look like the complex opposite the fundamental chamber inside a spherical building, which is a key tool in showing expansion.
The talk will cover joint work with Laura Grave de Peralta and Izhar Oppenheim.