Implicitization and Syzygies  

Given a rational map between projective spaces, a natural question that arises is how to study the (closed) image as a subvariety. As the coordinates are given parametrically, a natural question that arises is how to obtain the corresponding implicit system. This so-called implicitization problem has been studied to great length by geometers, algebraists and, in recent years, the geometric modeling community, for its applications to computer-aided design. In this talk, we discuss the connections of this geometric problem to the algebraic notion of syzygies. We discuss the history of the problem, as well as its modern treatment and applications, and end with a variety of open-ended questions for future research.