A mathematical structure $M$ is homogeneous if every isomorphism between its ``small'' (typically: finite, or finitely generated) substructures extends to an automorphism of $M$. Fraisse theory provides tools for constructing many examples of homogeneous structures. We will discuss a stronger notion, called uniform homogeneity, which leads to embeddings of the automorphism groups of small substructures of $M$ into the automorphism group of $M$. As it happens, typical homogeneous structures are uniformly homogeneous. We shall present examples of homogeneous structures, showing that uniform homogeneity is significantly stronger than homogeneity.

The talk is based on a joint work with S. Shelah (\texttt{https://arxiv.org/abs/1811.09650}).