Title: Expanders from local to global

Abstract: Imagine a network—like a social network, a transportation system, or even a biological system—where every part of the network is robustly connected to the rest. Expander graphs are the mathematical idealization of such networks. They are structures where any small group of points (nodes) has many connections to the rest of the graph, ensuring that no part is isolated and information (or influence) spreads efficiently throughout.
We will begin by surveying expander graphs, their discovery and construction, and some fascinating applications such as error-correcting codes, pseudorandomness, and probabilistically checkable proofs (PCPs), highlighting their role as a foundation for many breakthroughs in theoretical computer science. Then, we will shift focus to an exciting new kind of expanders called high dimensional expanders (HDXs). While expanders are well-understood and widely applied, HDXs remain enigmatic, with potential that we are only starting to uncover. We will talk about a fascinating local to global feature that HDXs have, and some applications.