Perfect and Scattered Subsets of Generalized Cantor Space VI

Speaker: 

Geoff Galgon

Institution: 

UCI

Time: 

Monday, November 16, 2015 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

We continue our discussion of perfect and scattered subsets in the generalized Cantor space. We we continue our study of the collection of topologies over 2^{\lambda} introduced last time. These topologies rely on the notion of a P_{\kappa}\lambda-forest, which is a natural generalization of a tree.

Perfect and Scattered Subsets of Generalized Cantor Space V

Speaker: 

Geoff Galgon

Institution: 

UCI

Time: 

Monday, November 9, 2015 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

We continue our discussion of perfect and scattered subsets in the generalized Cantor space. This week we finish the proof of the fact that \kappa-closed forcings don't add branches to \kappa-scattered subsets of 2^{\kappa}. We then introduce a collection of topologies over 2^{\lambda} whose restrictions to P_{\kappa}\lambda have some desirable properties. These topologies will rely on the notion of a P_{\kappa}\lambda-forest, which is a natural generalization of a tree. 

Perfect and Scattered Subsets of Generalized Cantor Space IV

Speaker: 

Geoff Galgon

Institution: 

UCI

Time: 

Monday, November 2, 2015 - 4:00pm to 5:30pm

We continue our discussion of perfect and scattered subsets in the generalized Cantor space. We focus this week on generalizing the games played on subsets of 2^{\omega} considered previously to the 2^{\kappa} context, and introduce alternate notions of \kappa-perfect and \kappa-scattered. We show that \kappa-closed forcings can’t add branches to \kappa-scattered subsets of 2^{\kappa} if \kappa isn’t a strong limit, which has as an immediate corollary the well-known lemma of Silver which says that \kappa-closed forcings can’t add branches to \kappa-trees (again for \kappa not a strong limit).

Perfect and Scattered Subsets of Generalized Cantor Space III

Speaker: 

Geoff Galgon

Institution: 

UCI

Time: 

Monday, October 26, 2015 - 4:00pm to 5:30pm

We continue our discussion of perfect and scattered subsets in the generalized Cantor space. We focus in particular this week on constructing certain types of trees in 2^{<\kappa} for uncountable \kappa which exhibit fundamentally different behavior than trees in 2^{<\omega} can, from the perspective of adding branches, cardinal dichotomies, etc. We also generalize the games previously discussed, and introduce alternative notions of \kappa-perfect and \kappa-scattered.

Perfect and Scattered Subsets of Generalized Cantor Space II

Speaker: 

Geoff Galgon

Institution: 

UCI

Time: 

Monday, October 19, 2015 - 4:00pm to 5:30pm

Location: 

RH 440R

We will initially discuss games played on subsets of the Cantor space, for which the existence or nonexistence of winning strategies for certain players can provide a characterization of perfectness or scatteredness. We will also give an old characterization of the type of trees in 2^{<\omega} through which outer models can add branches. Finally, we will make some observations about the nature of some generalizations of these topics to the 2^{\kappa} spaces.

Perfect and Scattered Subsets of Generalized Cantor Space

Speaker: 

Geoff Galgon

Institution: 

UCI

Time: 

Monday, October 12, 2015 - 4:00pm to 5:30pm

Location: 

440R

We will recall the standard notions of perfect and scattered subsets of 2^{\omega} and give several equivalent characterizations, make some observations, and record some facts. We will then discuss how some natural analogues to these characterizations diverge from one another in the generalized 2^{\kappa} setting. We will make some observations and give some open questions/directions.

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