Title:  Constructing ideals with large projective dimension

Abstract:  Projective dimension is a homological measure of the complexity of algebraic objects. Motivated by computational considerations, M. Stillman asked for upper bounds on the projective dimension of homogeneous ideals in polynomial rings, based solely on invariants of the ideal, not of the ambient ring. In this talk, we discuss several constructions that shed some light on what ideal invariants can or cannot be used to bound projective dimension and we give  lower bounds on any possible answer to Stillman's question. The talks is based on joint work with Huneke-Mantero-McCullough and Beder-McCullough-Nunez-Snapp-Stone.