It has long been known that the automorphism group of a countable first-order
structure is (isomorphic to) a closed subgroup of S_infty, and conversely
any such group is(isomorphic to) the automorphism group of some countable
first order structure (which can also assumed to be relational and
ultrahomogeneous). This has led to an interesting interplay between model
theory and the descriptive theory of actions of subgroups of S_infty.
In this talk, we will explain how the concept of metric structure (as
introduced by Ben Yaacov, Berenstein, Henson and Usvyastov) leads to a
similar interplay between so-called continuous logic and the descriptive
theory of Polish groups and their actions.
We will in particular explain why this leads to an extension of the concept of ample generics (introduced by Kechris and Rosendal) and present examples and applications of this new concept. Finally, if time allows, we will discuss some related open questions.