The method of matrix iterations was introduced by Blass and Shelah in their study of the dominating and the ultrafilter numbers. Since its appearance, the method has undergone significant development and applied to the study of many other cardinal characteristics of the continuum, including those associated to measure and category.

Recently, we were able to extend the technique of matrix iterations to a “third dimension” and so, evaluate the almost disjointness number in models where previously its value was not known. In addition, we obtain new constellations of the Cichon diagram (with up to seven distinct values). This is a joint work with Friedman, Mejia and Montoya.

The bounding, splitting and almost disjoint families are some of the well studied infinitary combinatorial objects on the real line. Their study has prompted the development of many interesting forcing techniques. Among those are the method of creature forcing, as well as Shelah’s template iteration techniques.

In this talk, we will discuss some recent developments of Shelah’s template iteration methods, leading to models in which the bounding, the splitting and the almost disjointness numbers can be quite arbitrary. We will conclude with a brief discussion of open problems.

Slides

The Riemann rearrangement theorem states that a convergent real series $sum_n a_n$ is absolutely convergent if and only if the value of the sum is invariant under all rearrangements $sum_n a_{p(n)}$ by any permutation $p$ on the natural numbers; furthermore, if a series is merely conditionally convergent, then one may find rearrangements for which the new sum $sum_n a_{p(n)}$ has any desired (extended) real value or which becomes non-convergent. In recent joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson, based on an exchange in reply to a Hardy’s MathOverflow question on the topic, we investigate the minimal size of a family of permutations that can be used in this manner to test an arbitrary convergent series for absolute convergence.  Specifically, we define the rearrangement number $rr$, a new cardinal characteristic of the continuum, to be the smallest cardinality of a set $P$ of permutations of the natural numbers, such that if a convergent real series $sum_n a_n$ remains convergent to the same value after any rearrangement $sum_n a_{p(n)}$ by a permutation $p$ in $P$, then it is absolutely convergent. The corresponding rearrangement number for sums, denoted rr_Sigma, is the smallest cardinality of a family $P$ of permutations, such that if a series $sum_n a_n$ is conditionally convergent, then there is some rearrangement $sum_n a_{p(n)}$, by a permutation $p$ in $P$, for which the series converges to a different value. We investigate the basic properties of these numbers, and explore their relations with other cardinal characteristics of the continuum. Our main results are that b≤ rr≤ non(M), that d≤ rr_Sigma, and that b≤ rr is relatively consistent.

MathOverflow question | JDH blog post about this talk

In a throw-away comment in a relatively recent preprint, Garti and Shelah state that using the technique of Dzamonja and Shelah, one can start with a model of set theory containing a supercompact cardinal κ, and force to obtain a model in which κ remains supercompact, 2^κ is large, but the ultrafilter number at κ is only κ⁺. I will present this construction, and with it further results from joint work with Vera Fischer and Diana Montoya pinning down many other generalized cardinal characteristics at κ in the resulting model.