Jörg Brendle: Cardinal invariants of analytic quotients

Week 1, Tuesday June 16, 10:05-10:55

Let I be a definable (e.g., an analytic) ideal on the natural numbers omega, and consider the quotient P(omega)/I, equipped with the ordering induced by the inclusion relation on omega. If I = Fin is the ideal of finite sets, this structure has been intensively investigated, and a number of cardinal invariants which describe its combinatorial properties have been defined.
In this talk I will present a few results on analogous cardinal invariants for the case where I is an ideal distinct from Fin. If I is not F_sigma, the quotient is not sigma-closed in general, and some cardinal invariants may become countable. Therefore I will concentrate on the case where I is an F_sigma-ideal. It turns out that P(omega)/I looks rather similar to P(omega)/Fin in this case, that many inequalities between cardinal invariants which hold in the classical case can be generalized, and similarly for consistency results. In fact, cardinal invariants of P(omega)/I are rather hard to distinguish from their classical counterparts. However, we know a few consistency results. For example, the splitting number of P(omega)/I where I is any summable ideal may be strictly smaller than the splitting number of P(omega)/Fin.