Global \(\Sigma\)-uniformization is a striking regularity property for projective sets of reals. Until recently, the only known method for obtaining it relied on the existence of a good projectively definable well-ordering of the reals. Consequently, the global \(\Sigma\)-uniformization property appeared to be confined to a rather limited class of universes.

In this talk, I will present a forcing technique that yields models satisfying global \(\Sigma\)-uniformization. The method is highly flexible and can be adapted to produce a wide range of universes in which global \(\Sigma\)-uniformization coexists with other desirable or surprising features. In particular, I will show how to force \(\mathsf{BPFA}\) together with global \(\Sigma\)-uniformization—a result that contrasts sharply with the situation under \(\mathsf{PFA}\), which implies \(\mathsf{PD}\) and thus the familiar zig-zag pattern governing the uniformization property.

This is the second part of a series of 3 talks; the first part has been on November 13 and the next and last talk will take place on December 2.