In this three-part lecture series, I will present my recent result with Yair Hayut that it is consistent for all successors of regular cardinals to carry dense ideals.
We will start a bit out of order with applications, beginning with Woodin’s "transfer theorem" that shows that if we have diamonds and a normal ideal \(J\) on \(\kappa^+\) such that \(\mathcal{P}(\kappa^+)/J\) is equivalent to \(\mathrm{Col}(\kappa\), \(\kappa^+\)), then there is a uniform \(\kappa\)-complete ideal \(K\) on \(\kappa^+\) such that \(\mathcal{P}(\kappa^+)/K\) is isomorphic to \(\mathcal{P}(\kappa)/\mathrm{bounded}\). From this we can derive several combinatorial consequences that address some questions from graph theory and recent work on homological algebra on the ordinals.
In the second and third lecture, we will outline the consistency proof.