Fri Sep 15, 2023
Scott Mutchnik (University of Illinois Chicago)
Perspectives on NSOP3
Since at least the first decade of the 21st century, two questions have troubled model theorists: how can we extend (neo)stability-theoretic methods beyond simplicity (and now, beyond NSOP2), and is NSOP2 equal to NSOP3? Recent progress on the equality of NSOP1 and NSOP2 has offered hope that these two questions are related. However, while new stability-theoretic relations such as Conant-independence and n-ð-independence have proven promising in making sense of the higher NSOPn hierarchy, positive global consequences of NSOPn, for n > 2, have continued to elude us. We discuss our recent finding of the first such results, on NSOP3. While it is still open whether NSOP3 coincides with NSOP2 (i.e. with NSOP1), these results are concrete in that they do not, in general, hold in NSOP4 theories. Previously, NSOP3 has been thought of as very different from NTP2, in the sense that there is no known NSOP3 NTP2 theory which is not also simple. It is therefore surprising that our results on NSOP3 theories illustrate similar behavior to NTP2 theories.