The uniformization property, introduced by N. Lusin in 1930, is an extensively studied notion in descriptive set theory. For a given projective pointclass \(\Gamma\) it says that every subset of the plane which belongs to \(\Gamma\) has a uniformizing function whose graph is an element of \(\Gamma\) as well. The celebrated results of Y. Moschovakis on the one hand and D. Martin, J. Steel and H. Woodin on the other, yield a natural and global description of the behaviour of the uniformization property for projective pointclasses under the assumption of large cardinals. In particular, under PD, for every natural number \(n\), \(\Pi^1_{2n+1}\)-sets and hence \(\Sigma^1_{2n+2}\)-sets do have the uniformization property.

Yet the question of universes which display an alternative behaviour of theses regularity properties has remained in large parts a complete mystery, mostly due to the absence of forcing techniques to produce such models. Consequentially, a lot of very natural problems have remained wide open ever since.

In my talk, I want to outline some recently obtained tools, which turn the question of forcing a universe with the \(\Pi^1_n\)-uniformization property into a fixed point problem for certain sets of forcing notions. This fixed point problem can be solved, yielding a specific set of forcing notions which in turn can be used to force the Uniformization property (for \(n>2\)) over fine structural inner models with large cardinals (for \(n=3\), the inner model is just L). For even $n$, these universes witness for the first time the consistency (relative to the existence of \(n-3\) many Woodin cardinals) of the \(\Pi^1_{n}\)-uniformization property, and, for odd \(n\), give new lower bounds in terms of consistency strength.

The talk will be held in a hybrid mode via Zoom. Students at Uni Wien are strongly encouraged to attend the seminar in person.