## Logic Seminar

Scott Mutchnik

University of Illinois Chicago

Perspectives on NSOP_3

**Abstract:**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 NSOP_2), and is NSOP_2 equal to NSOP_3? Recent progress on the equality of NSOP_1 and NSOP_2 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 NSOP_n hierarchy, positive global consequences of NSOP_n, for n > 2, have continued to elude us. We discuss our recent finding of the first such results, on NSOP_3. While it is still open whether NSOP_3 coincides with NSOP_2 (i.e. with NSOP_1), these results are concrete in that they do not, in general, hold in NSOP_4 theories. Previously, NSOP_3 has been thought of as very different from NTP_2, in the sense that there is no known NSOP_3 NTP_2 theory which is not also simple. It is therefore surprising that our results on NSOP_3 theories illustrate similar behavior to NTP_2 theories.

Tuesday September 12, 2023 at 4:00 PM in 636 SEO