Shelah defined the space $S_{Sh}(T)$ of strong types of a first order theory $T$. A profinite Galois group acts on this space; the quotient is the space of types $S(T)$. For a stable theory, Shelah showed that $S_{Sh}$ is the gateway to all analysis of interactions between substructures of models of $T$.

Lascar in 1982 undertook a generalization to arbitrary theories. As later clarified, the space $S_{KP}$ of Kim-Pillay types can be defined; a compact group $G^*$ acts on $S_{KP}$, with quotient $S(T)$. For simple theories, or ones with a definable measure, an excellent theory of amalgamation becomes possible.

For arbitrary theories, the investigation continued at the hands of Newelski, Krupinski, Rzepecki, Pillay, and others; a new compact group was found by means of a journey to topological dynamics. I will recount some of this story, and describe a more finitary compact group, that arises as the automorphism group of a new space generalizing $S_{KP}$, the pattern space of $T$. The elements of the quotient space here are not the types but the maximal elements of the Lascar-Poizat fundamental order.