Cofinalities of the symmetric group and of the ultrapowers of ω

We consider the following cardinal invariants of the continuum: cf(Sym(ω)), being the cofinality of the group of permutations of ω, and mcf, defined as the minimal cofinality of an ultrapower of ω over any non-principal ultrafilter. During the talk we will present known results regarding the relation between those two invariants and some other cardinal invariants of the continuum, such as b, d and g. Some results from the fairly recent paper by H. Mildenbergen and S. Shelah (namely "The Cofinality of the Symmetric Group and the Cofinality of Ultrapowers") will also be presented, such as the relative consistency of ZFC + "cf(Sym(ω)) > mcf" or that of ZFC + " b <= cf(Sym(ω)) < mcf".