Explore star clusters in clique complexes and the Vietoris-Rips complex of planar sets in this 50-minute conference talk by Jonathan Barmak. Delve into the concept of star clusters in simplicial complexes and their contractibility in clique complexes. Discover applications of this notion in studying homotopy invariants of independence complexes of graphs. Examine the Vietoris-Rips complex of planar sets, including its free fundamental group property and the conjecture about its homotopy type. Learn how star clusters can be utilized to obtain homotopy information of the Vietoris-Rips complex. The talk covers topics such as independence complexes of graphs, proofs, applications, nontrivial lower bounds, contractile subcomplexes, and concludes with questions and discussions on uniform bounds.