Cardinal characteristics on the generalized Baire and Cantor spaces $\kappa^\kappa$ and $2^\kappa$ have recently generated significant interest. In this talk I will introduce a different generalization of cardinal characteristics, namely to the space of functions $f:\omega^\omega \to \omega^\omega$. Given an ideal $I$ on Baire space and a relation $R$ let us define $f R_I g$ for $f$ and $g$ functions from $\omega^\omega$ to $\omega^\omega$ if and only if $f(x) R g(x)$ for an $I$-measure one set of $x \in \omega^\omega$. By letting $I$ vary over the null ideal, the meager ideal and the bounded ideal; and $R$ vary over the relations $\leq^*$, $\neq^*$ and $\in^*$ we get 18 new cardinal characteristics by considering the bounding and dominating numbers for these relations. These new cardinals form a diagram of provable implications similar to the Cichoń diagram. They also interact in several surprising ways with the cardinal characteristics on $\omega$. For instance, they can be arbitrarily large in models of CH, yet they can be $\aleph_1$ in models where the continuum is arbitrarily large. They are bigger in the Sacks model than the Cohen model. I will introduce these cardinals, show some of the provable implications and discuss what is known about consistent inequalities, focusing on the $\mathfrak{b}$-numbers in well-known models such as the Cohen and Random model. This is joint work with Jörg Brendle.

Bio: Corey Bacal Switzer is currently a postdoctoral researcher at the Kurt Gödel Research Center For Mathematical Logic in the Mathematics Department of the University of Vienna working under Vera Fischer. He finished his PhD at the CUNY Graduate Center in New York in 2020. His research is in set theory, focusing on forcing, cardinal characteristics and infinite combinatorics.