I shall speak about the correspondence between (locally) countably categorical structures (which are mathematical structures that are particularly easy to describe using formal logic) and their automorphism groups, the (locally) Roelcke precompact Polish groups. The non-local correspondence is by now folklore, and lies at the origin of several interactions between model theory and topological dynamics. Locally Roelcke precompact groups were studied by Zielinski, while local categoricity is close in spirit to Hrushovski's "local logic". With Todor Tsankov we recently showed that these two notions fit together perfectly in an extension of the aforementioned correspondence.