Time:  12  1:20 p.m. 
Room: 
CFA 110

Speaker: 
Natasha Dobrinen Chowla Research Assistant Professor Department of Mathematics Pennsylvania State University 
Title: 
The hyperweak distributive law and related infinitary games in
Boolean algebras

Abstract: 
The work we will present is joint with James Cummings. The
hyperweak distributive law in Boolean algebras, invented by Prikry, is a
nontrivial generalization of the threeparameter distributive law. It
fails in the Cohen algebra, but holds in many other Boolean algebras. We
will define the hyperweak distributive law and a related infinitary
twoplayer game in Boolean algebras, and show some implications between
the existance or nonexistance of a winning strategy for either player and
the hyperweak distributive law.
We will also show that it is consistent with ZFC that for all infinite
cardinals $\kappa$, for each infinite regular carinal $\nu \le \kappa$
there is a $\kappa^+$Suslin algebra containing a $<\nu$closed dense
subset in which many games of length $\ge \nu$ are all undetermined. To do
this, we use $\square_{\kappa}$ and $\diamond_{\kappa^+}(S)$ for all
stationary sets $S$ contained in $\kappa^+$. This improves on an earlier
result of Dobrinen (03) which showed that for regular cardinals $\kappa$,
there is consistently a large gap between the strengths of "$B$ satisfies
the $(\kappa,\infty)$d.l." and "player II has a winning strategy in the
game $\mathcal{G}^{\kappa}_1(\infty)$.

Organizer's note:  