7 problems

Let a pseudofinite group be a group satisfying the theory of all finite groups, and let its Bohr compactification be its universal compactification, namely a compact group receivin…

Let $G$ be a countably infinite group that is both omega-categorical, meaning that its theory has a unique countable model up to isomorphism, and pseudofinite, meaning that it is i…

Let $G$ be an infinite simple group of finite Morley rank, and let $\alpha$ be a generic automorphism of $G$. The fixed-point subgroup of $\alpha$ is $C_G(\alpha)$. Principal conje…

Commutativity conjecture. If $G$ is a pseudofinite group, then $(bG)^{0}$ is commutative.

Felgner's conjecture. Every simple pseudofinite group is isomorphic to a twisted or untwisted Chevalley group over a pseudofinite field.

Fixed-point implication conjecture. If the group of fixed points of $\alpha$ is pseudofinite, then $G$ is isomorphic to a Chevalley group over an algebraically closed field.

Hrushovski's fixed-point conjecture. The group of fixed points of $\alpha$ is pseudofinite, that is, an infinite model of the theory of finite groups.