Exceptional sets of orthogonal projections: Classical results and current research

The projection problem of geometric measure theory is the following: given a set A ⊆ ℝn, what is the relationship between the geometric properties of A and those of its projections onto k-dimensional subspaces of ℝn? One way of answering this is to bound the dimension of the so-called exceptional set of orthogonal projections of A—the set of k-planes onto which the orthogonal projection of A has unusually low dimension compared to the dimension of A. Our purpose is to review the most important results of this sort, with an eye toward the technical tools involved in their proof, and to survey the author's own research in this area.