# Forcing and the Separation, the Reduction and the Uniformization-property.

The Separation property, the Reduction property and the Uniformization property, introduced in the 1920's and 1930's are three classical regularity properties of pointclasses on the reals. The celebrated results of Y. Moschovakis on the one hand and D. Martin, J. Steel and H. Woodin on the other, yield a global description of the behaviour of these regularity properties 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 (and therefore the weaker Reduction property and the Separation property for the dual pointclass).

Yet the question of universes which display an alternative behaviour of theses regularity properties has remained a complete mystery, mostly due to the absence of forcing techniques to produce such models. Indeed, even the question of the forceability of a universe where the $\Sigma^1_3$ Separation property holds was a well-known open problem since 1968.

In my talk, I want to outline some recently obtained techniques, which turn the question of a universe with, say, the $\Pi^1_3$ Reduction 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 $\Pi^1_n$ Reduction property or, with more complicated techniques, the $\Pi^1_n$ 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 outright contradict the PD-induced pattern, for odd n these universes give new lower bounds in terms of consistency strength.