PhD Defense Karen Bakke Haga

Title: MAXIMAL ALMOST DISJOINT FAMILIES,DETERMINACY, AND FORCING

This thesis is based on the paper Maximal almost disjoint families, determinacy, and forcing, 2018, which is joint work with David Schrittesser and Asger Törnquist.

We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a Borel ideal on $\omega$. We show that if $\mathcal J$ is an arbitrary
$F_\sigma$ ideal, or is any finite or countably iterated Fubini product of $F_\sigma$ ideals, then there are no analytic infinite $\mathcal J$-MAD families; and assuming Projective Determinacy there are no infinite projective $\mathcal J$-MAD families; and under the full Axiom of Determinacy + $V=\eL(\R)$ there are no infinite $\mathcal J$-mad families. These results apply in particular when $\mathcal J$ is the ideal of finite sets $\fin$, which corresponds to the classical notion of MAD families. The proofs combine ideas from invariant descriptive set theory and forcing.

Supervisor: Asger Dag Törnquist, MATH, University of Copenhagen

Assessment committee:

Chairman: Prof. Ryszard Nest,  MATH, University of Copenhagen
Prof. Mirna Dzamonja, University of Norwich, UK
Prof. William Weiss, University of Toronto