The Samuel compactification, or the greatest ambit, is an important

compactification of a topological group for its dynamics. In the case of discrete groups, the Samuel compactification coincides with the \v{C}ech-Stone compactification and its algebra and combinatorics have been extensively studied. We remind the Samuel compactification for automorphism groups in the ultrafilter

language and point out some differences and similarities with the discrete case. We will then apply algebra and combinatorics to answer a problem of Ellis for the group of permutations of the integers. This is a joint work in progress with Andy Zucker (Carnegie Mellon University).