A nontrivial uniform algebra Dirichlet on its maximal ideal space

It is shown that there exists a nontrivial uniform algebra that is Dirichlet on its maximal ideal space and has a dense set of elements that are exponentials. This answers a 65-year-old question of John Wermer and a 17-year-old question of Garth Dales and Joel Feinstein. Our example is P(X) for a certain compact set X in complex Euclidean 2-space (${\mathbb{C}}^2$). It is also shown that there exists a logmodular uniform algebra with proper Shilov boundary but with no nontrivial Gleason parts. This answers a modification of another 65-year-old question of Wermer.