"Local-to-Global" Theorems on High Dimensional Expanders

Expansion in graphs is a well studied topic, with a wealth of applications in many areas of mathematics and the theory of computation.

High dimensional expansion is a generalization of expansion from graphs to higher dimensional objects, such as simplicial complexes or partially ordered sets.

High dimensional expanders are still far from understood, but one fascinating new aspect is how global properties emerge from local ones. This phenomenon has already led to progress on longstanding questions in the areas of error-correcting codes, and Probabilistically Checkable Proofs (PCPs).

I will describe the two key definitions: cosystolic expansion and link expansion. I will then describe how these relate to some of the exciting new applications.