On sequences of homomorphisms into measure algebras and the Efimov problem

The starting point for my talk, based on the joint work with Piotr
Borodulin-Nadzieja, is our theorem presented by him recently at this
seminar, characterizing a special class of compact spaces without
convergent sequences in the random model. Namely, we proved that if A
is a Boolean algebra in the ground model V and M a measure algebra,
then the Stone space of A has no non-trivial convergent sequences in
the random extension V^M if and only if the family H(A,M) of all
homomorphisms from A to M satisfies in V some special sequential
property, expressed in terms of various topologies on H(A,M). During
my talk I'll describe exactly those topologies, show some relations
and differences between them, connect them with well-known topologies
on spaces of measures, as well as state formally the aforementioned
property and present the relation between the theorem and the famous
Efimov problem.