Veranstaltungsübersicht -- Sommersemester 2023

A forcing order is said to be $<\kappa$-distributive iff it does not add new sequences of length $<\kappa$. A sufficient but not necessary condition for this is that the forcing is $<\kappa$-closed, i.e. any $<\kappa$-sequence of conditions has a lower bound. We introduce a strenghtening of $<\kappa$-distributivity called strong $<\kappa$-distributivity which can replace $<\kappa$-closure in many applications. A main benefit of this property is that a $<\kappa$-closed forcing remains strongly $<\kappa$-distributive in any extension by a $\kappa$-cc. order, even though it no longer necessarily is $<\kappa$-closed.

A theory is noetherian if there is a family of definable sets with the descending chain condition such that every definable set is a boolean combination of those in the family. Noetherianity captures some of the desired properties of algebraically closed fields in any characteristic or differentially closed fields in characteristic 0. Noetherian theories are in particular omega-stable and equational. In recent work with M. Ziegler, we have shown that the theory of proper pairs of algebraically closed fields in any characteristic is noetherian.

It is a well-known fact, with important applications in additive combinatorics and Ramsey theory, that the usual sum of integers may be extended to the space of ultrafilters over Z, yielding a compact right topological semigroup. The analogous construction also goes through for the product.

Recently, B. Šobot introduced two (ternary) notions of congruence on the space above. I will talk about joint work with M. Di Nasso, L. Luperi Baglini, M. Pierobon and M. Ragosta, in which the study of these congruences led us to isolate a class of ultrafilters enjoying characterisations in terms of tensor products, directed sets, profinite groups, and more.