Both the automorphism group \(\operatorname{Aut}(M)\) and the monoid of elementary embeddings \(\operatorname{EEmb}(M)\) of a structure \(M\) are naturally endowed with a topology known as the topology of pointwise convergence. A large body of work in model theory, group theory, and semigroup theory is dedicated to the problem of recovering this topology from the purely algebraic behaviour of the associated space of symmetries, with a focus on \(\omega\)-categorical structures (i.e., structures whose first-order theory has a unique countable model up to isomorphism). It turns out that the topology of pointwise convergence behaves very differently on automorphism groups and monoids of elementary embeddings. In particular, recently, Pinsker and Schindler showed that the topology of pointwise convergence has a purely algebraic description on the monoids of elementary embeddings of \(\omega\)-categorical structures with trivial algebraic closure: they show that the topology of pointwise convergence coincides with the Zariski topology, whose generating open sets are solutions to semigroup inequalities. This is also used to show that the topology of pointwise convergence is minimal amongst Hausdorff semigroup topologies, in the sense that it is the coarsest Hausdorff semigroup topology on \(\operatorname{EEmb}(M)\). We show that instead if \(\operatorname{Aut}(M)\) has non-trivial centre, the Zariski topology is never Hausdorff on \(\operatorname{EEmb}(M)\), and so does not agree with the topology of pointwise convergence. Nevertheless, adapting previous work of de la Nuez Gonzales and Ghadernezhad on automorphism groups, we show that the topology of pointwise convergence is always minimal amongst \(T_1\) semigroup topologies for a large class of simple \(\omega\)-categorical structures.

This talk is based on ongoing work with de la Nuez Gonzales, Ghadernezhad, and Pinsker.