I will discuss recent investigations of various reducibility notions between Pi^1_2 principles of second-order arithmetic, the most familiar of which is implication over the subsystem RCA_0. In many cases, such an implication is actually due to a considerably stronger reduction holding, such as a uniform (a.k.a. Weihrauch) reduction. (Here, we say a principle P is uniformly reducible to a principle Q if there are fixed reduction procedures Phi and Gamma such that for every instance A of P, Phi(A) is an instance of Q, and for every solution S to Phi(A), Gamma(A + S) is a solution to A.) As an example, nearly all the implications between principles lying below Ramsey's theorem for pairs are uniform reductions. In general, the study of when such stronger implications hold and when they do not gives a finer way of calibrating the relative strength of mathematical propositions, and has led to the development of a number of new forcing techniques for constructing models of second-order arithmetic with prescribed combinatorial properties. In addition, this analysis sheds light on several open questions from reverse mathematics, including that of whether the stable form of Ramsey's theorem for pairs (SRT^2_2) implies the cohesive principle (COH) in \omega (standard) models of RCA_0.

The pursuit of better understanding the universe of set theory V motivated an extensive study of definable inner models M whose goal is to serve as good approximations to V. A common property of these inner models is that they are contained in HOD, the universe of hereditarily ordinal definable sets. Motivated by the question of how ``close" HOD is to V, we consider various related forcing methods and survey known and new results. This is a joint work with Spencer Unger.

We complete the proof of the main theorem by showing that if X^3 is isomorphic to X, then X^{\omega} has a parity-reversing automorphism. By our previous results this implies X^2 is isomorphic to X as well. The proof generalizes to show that for any n > 1, if X^n is isomorphic to X, then X^2 is isomorphic to X. Time permitting we will discuss related results, including the existence of an A and X such that A^2X is isomorphic to X, while AX is not.

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.