Given a graph \(G\), its independence complex is the simplicial complex whose vertex set is \(V(G)\) and whose faces are precisely the independent sets of \(G\). In this talk, we will take a panoramic look at the independence complex, beginning with its construction and developing an intuition for why this seemingly simple object has attracted considerable interest. We will then explore its significance through several examples and results from the literature. In particular, we will see how independence complexes naturally appear in different combinatorial and topological settings.
Finally, we will discuss how independence complexes arise in our own work and how they provide a useful tool for studying the topology of Vietoris-Rips complexes. The goal of the talk is to give a gentle introduction to independence complexes and to showcase some of the interesting mathematics that emerges from this simple construction.
