It was once conjectured that whenever a compact set in complex Euclidean space has a nontrivial polynomial hull, there must be an analytic disc in the hull. This conjecture was disproved by Stolzenberg in 1963. Nevertheless, attempts to prove the conjecture led to much interesting mathematics. In particular, Gleason's introduction of his parts was motivated by the conjecture. Garnett proved in 1967 that in no reasonable sense can Gleason parts be regarded as analytic sets in general. However, the uniform algebras constructed by Garnett do have a great deal of analytic structure. I will present recent joint work with Papathanasiou that strengthens Garnett's result by obtaining the Gleason parts in the complete absence of analytic discs. Speaker(s): Alex Izzo (Bowling Green State University)