Notions of Higher Type

Abstract: Notions of finite type play an important role in several complex variables. The most standard notion is D'Angelo type, which measures the order of contact of holomorphic curves with the boundary of a domain in ${\mathbb C}<sup>n$. For the $\bar \partial$-Neumann problem, however, the order of contact of the boundary of the domain with $q$-dimensional complex varieties controls its behavior on $(p,q)$ forms. There are two different ways of measuring this order of contact, one due to John D'Angelo and another due to David Catlin. We survey the known results about the D'Angelo and Catlin $q$-types, their relationship, and other notions that complete the picture.