A semigroup is a set together with an associative binary operation. As opposed to stable groups, the model theory of stable semigroups is not so rich. One reason for that is their abundance. We will review (and prove) some known results on type-definable semigroups in stable structures and offer some examples and counter-examples.
Uniform definability of types over finite sets (UDTFS) is a property of formulas which implies NIP and characterizes NIP in the level of theories (by Chernikov and Simon).
In this talk we will prove that if T is any theory with definable Skolem functions, then every dependent formula phi has UDTFS. This result can be seen as a translation of a result of Shay Moran and Amir Yehudayof in machine learning theory to the logical framework.
G-compactness, hereditary G-compactness and related phenomena
The notion of G-compactness, along with the Galois groups, was introduced by Lascar in order to find a sufficient condition under which a first order theory can be recovered from the category of its models. I will recall this notion. In order to do that, I will also recall various classical notions of strong types, and possibly the Galois group of the theory (and briefly discuss their importance).