  We study problems concerning the Samuel compactification of the automorphism
group of a countable first-order structure. A key motivating question is a
problem of Furstenberg and a counter-conjecture by Pestov regarding the
difference between $S(G)$, the Samuel compactification, and $E(M(G))$, the
enveloping semigroup of the universal minimal flow. We resolve Furstenberg's
problem for several automorphism groups and give a detailed study in the case
of $G = S_\infty$, leading us to define and investigate several new types of
ultrafilter on a countable set.
