Abstract: 

The Grothendieck group of a semigroup is the universal homomorphism from that semigroup into a group. In particular, if the semigroup is a compact left topological semigroup, we can ask about the relationship between its Grothendieck group and its Ellis groups. It turns out that the Grothendieck group is always a (typically nontrivial) quotient of the Ellis group. If we consider various semigroups of types in model theory, then the Grothedieck groups can be naturally found as factors of various standard homomorphisms for example the homomorphism from the Ellis group of finitely satisfiable types in a definable group to the quotient G/G^{000}. This gives rise to new variants of the Ellis group conjecture. In my talk, I will explain this in more detail, and, time permitting, discuss some examples disproving some variants of the above, as well as some criteria for when they do hold. This is joint work with Krzysztof Krupiński.