GRADUATE RESEARCH SEMINAR--Fall 2026

Tuesdays 2:00-3:00 PM

SEO 427

Office hours: Tue 1:00-2:00 PM and 4:00-5:00 PM, or by appointment, SEO 508

Syllabus

The following potential directions for discussion overlap at many points; which ones we will cover, and to what extent, will depend on the interests of the participants in this seminar. The larger categories are ordered thematically, rather than chronologically or by order of discussion in the seminar. For example, it may be worthwhile to discuss the development of the NSOP_2–NSOP_3 problem chronologically, building up to it through the semantic and syntactic techniques from the literature discovered by manual argument, before introducing Chernikov’s recent breakthrough using an elementary syntactic argument discovered using generative AI.

Syntax: The original definition of SOP_2 is a positive straight definition in the sense of Bailetti: it is defined by consistency and inconsistency conditions in a formula. It is also a classical result, essentially due to Shelah and Usvyatsov, based on Shelah’s work, that SOP_3 has a similar kind of definition, even if this is not immediate from Shelah’s original definition of SOP_n for integers n ≥ 3. This configuration is also isolated by Malliaris as the “compatible order property.” It can be used to show the non-immediate fact that SOP_3 implies SOP_2, as in Džamonja and Shelah. Moreover, resolving an open problem of Bailetti, with roots in a putative solution by Shelah, Day and Mutchnik show that SOP_n for integers n ≥ 4 satisfies the related condition of having a straight definition, defined by consistency and inconsistency conditions in a formula and its negation. Toward the end of the course we should discuss Chernikov’s fully syntactic proof that NSOP_2 is equal to NSOP_3, which uses generative AI. Readings: Shelah (1996), “Toward classifying unstable theories,” Definition 2.5 and Claim 2.8; Džamonja and Shelah (2004), “On ◁*-maximality,” Definition 2.2 and Claim 2.3; Shelah and Usvyatsov (2008), “More on SOP_1 and SOP_2,” Fact 1.3; Day and Mutchnik (2026), “On the notion of a patterning property in model theory”; Chernikov (2026), “SOP_2 = SOP_3.”

Supplementary readings: Shelah (1990), “Classification theory,” Theorem III.7.11; Shelah (1996), “Toward classifying unstable theories,” Theorem 2.9; Džamonja and Shelah (2004), “On ◁*-maximality”; Malliaris and Shelah (2016), “Cofinality spectrum problems in model theory, set theory and general topology”; Kaplan, Ramsey and Simon (2024), “Generic stability independence and treeless theories”; Bailetti (2024), “A walk on the wild side: Notions of maximality in first-order theories.”

Semantics, and its interaction with the syntactic: The proof that NSOP_1 is equal to NSOP_2 starts with proving a version of Kim’s lemma in NSOP_2 theories, and continues by proving that Conant-independence is symmetric in NSOP_2 theories. The proof is completed by showing that any theory with these two properties must have SOP_3 or be NSOP_1, using an argument of Conant that has also been applied by Mutchnik to make the problem of whether NSOP_2 is equal to NSOP_3 more tractable. Also of interest are the connections between NSOP_4 and Conant-independence, and between NSOP_{2^{n+1}+1} and n-ð-independence. Readings: Mutchnik (2025), “On NSOP_2 theories”; Conant (2017), “An axiomatic approach to free amalgamation,” Theorem 7.17; Mutchnik (2026), “Conant-independence and generalized free amalgamation”; Mutchnik (2023), “On the properties SOP_{2^{n+1}+1}.”

Supplementary readings: Mutchnik (2026), “Properties of independence in NSOP_3 theories”; Mutchnik (2023), “Generic expansions and the group configuration theorem.”

NSOP_r theories for non-integer values of r: A short but nontrivial argument is required to show that NSOP_r for real values r ≥ 3 is well-defined, in the sense that it coincides with Shelah’s original definition when r happens to be an integer. Once the real-valued NSOP_r hierarchy is established, this motivates a web of related combinatorial results: the integrality of 𝔬(ℋ) for ℋ a hereditary class defined by finitely many forbidden weakly embedded substructures, as well as progress toward the possible general non-integrality of 𝔬(ℋ), and ultimately the possible distinctness of the real-valued NSOP_r and integer-valued NSOP_n hierarchies, using interval helix maps. An important motivation for interval helix maps, part of the general framework of helix maps implicitly introduced by Malliaris, is that they are themselves powerful enough to prove the well-definedness and integrality results. Readings: Mutchnik (2025), “Approximations of the strict order property.”

Supplementary readings: Malliaris (2010), “Edge distribution and density in the characteristic sequence,” Section 7; Hanson (2020), “Definability and categoricity in continuous logic,” Appendix B.

Applications of the real-valued hierarchy to the integer-valued hierarchy: The real-valued NSOP_r hierarchy, the question of its distinctness from the integer-valued NSOP_n hierarchy, and the techniques arising from investigating this question have applications to apparently disparate questions on the integer-valued NSOP_n hierarchy. One approach to the originally open problem of whether NSOP_2 is equal to NSOP_3 is to develop the fine structure in between these properties. Toward this end, Mutchnik uses Kim-independence, specifically the theory introduced by Kaplan and Ramsey, to show that NSOP_r is well-defined for 2 < r < 3, an application of the equality of NSOP_1 and NSOP_2. Mutchnik also shows that one of two possibilities with different motivations must be the case: either the real-valued NSOP_r hierarchy does not coincide with the integer-valued NSOP_n hierarchy on sufficiently general grounds, or NSOP_{n+1} ∩ NTP_2 is equal to NSOP_n ∩ NTP_2 for integers n ≥ 3. Using techniques whose canonical setting is in the proof that 𝔬(ℋ) is an integer for ℋ a hereditary class defined by finitely many forbidden weakly embedded substructures (in a much more careful way!), Mutchnik also makes more unconditional progress on the then-open question of whether NSOP_2 is equal to NSOP_3: for every such hereditary class ℋ, if every theory whose models have age ℋ has SOP_2, then every theory whose models have age ℋ has SOP_3. This is not the case replacing “SOP_2” with “the tree property.” Given Chernikov’s proof that NSOP_2 is equal to NSOP_3, one goal of this discussion, besides demonstrating what it was possible to show by hand, should be to take stock of what additional information we get out of these results. For example, one consequence of the proof of the sharp dichotomy involving hereditary classes defined by finitely many forbidden weakly embedded substructures is an approximate implication NSOP_1 ⇝ NTP_2. Readings: Mutchnik (2026), “Some applications of the real strict order property hierarchy.”

Course schedule:

September 8: Introduction, SOP_3, SOP_2, and positive straight definability. (Džamonja and Shelah, Definition 2.2 and Claim 2.3; Shelah and Usvyatsov, Fact 1.3. See also Kaplan, Ramsey and Simon, Lemma 7.3 for a new characterization of SOP_3, leading to an excellent exposition of why SOP_2 implies SOP_3.)