A remarkable theorem of Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality, then there is a subset $x$ of $\kappa$, such that $HOD_x$ contains the powerset of $\kappa$. We show that in general this is not  the case for countable cofinality. Using a version of diagonal supercompact extender Prikry forcing, we construct a generic extension in which there is a singular cardinal $\kappa$ with countable cofinality, such that $\kappa^+$ is supercompact in $HOD_x$ for all $x\subset\kappa$. This result was obtained during a SQuaRE meeting at AIM and is joint with Cummings, Friedman, Magidor, and Rinot.

Building on our characterization from last week of the orders X that are isomorphic to AX, we characterize those X that are isomorphic to AAX. We then write down a condition -- namely, the existence of a parity-reversing automorphism (p.r.a.) for the countable power of A -- under which the implication ``AAX = X implies AX = X" holds. In future talks, we will show that if X is isomorphic to its cube then the countable power of X has a p.r.a., and hence X is isomorphic to its square.

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".