We continue toward a proof of the main theorem by characterizing, for a fixed linear order A, the collection of linear orders X such that AX is isomorphic to X.

In 1950's Sierpinski asked whether there exists a linear order X isomorphic to its lexicographicaly ordered cube but not to its square. We will give some historical context and begin the proof that the answer is negative. More generally, if X is isomorphic to any of its finite powers X^n (n>1) then X is isomorphic to all of them.

We continue the discussion of Viale-Weiss paper ``On the consistency strength of the proper forcing axiom".

Many consequences of the Proper Forcing Axiom (PFA) factor through the stationarity of the class of guessing models. Such consequences include the Tree Property at $\omega_2$, absence of (weak) Kurepa Trees on $\omega_1$, and failure of square principles.  On the other hand, stationarity of guessing models does not decide the value of the continuum, even when one requires that the guessing models are also indestructible in some sense.  I will give an introduction to the topic and discuss some recent results due to John Krueger and me.

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.