Perfect subsets of generalized Baire spaces and Banach-Mazur games
Mittwoch, 9.1.13, 16:30-17:30, Raum 404, Eckerstr. 1
Let \(\bkappa\) be an uncountable cardinal with \(\bkappa^{< \bkappa} = \bkappa\). We consider the generalized Baire space of functions \(f : \bkappa \bto \bkappa\) with basic open sets \(U_s = \b{f \bin \bkappa^\bkappa \bmid s \bsubseteq f \b}\) for \(s \bin {}^{< \bkappa} \bkappa\). A subset of \(\bkappa^\bkappa\) is perfect if it is the set of branches of a \(< \bkappa\)-closed subtree of \({}^{< \bkappa} \bkappa\) which splits above every node. We prove that after an inaccessible \(\blambda > \bkappa\) is collapsed to \(\bkappa^+\), every set \(A \bsubseteq \bkappa^\bkappa\) definable from ordinals and subsets of \(\bkappa\) either has size \(\bleq \bkappa\) or a perfect subset, and that the Banach-Mazur game for \(A\) is determined.
The Pila-Zannier proof of the Manin-Mumford conjecture
Mittwoch, 16.1.13, 16:30-17:30, Raum 404, Eckerstr. 1
Amoeba of Sacks forcing without Cohen reals
Mittwoch, 23.1.13, 16:30-17:30, Raum 404, Eckerstr. 1
On rings of continuous p-adic valued functions
Dienstag, 29.1.13, 11:30-12:30, Raum 318, Eckerstr. 1