GRADUATE RESEARCH SEMINAR--Fall 2025

2:00-3:00 PM

SEO 427

Office hours: Wed 2:00-3:00 PM, Fri 1:00-200 PM, 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. However, the very basics of the first direction will be key. Graduate students are encouraged to present.

The syntax-semantics correspondence: We will be interested in the characterization of simplicity in terms of the independence relation. We might also be interested in the characterization of NSOP_1 in terms of the Kim-independence relation, at least the statement of the results. To convey what the combinatorial content of NSOP_1 says at the level of forking, Kim's lemma in NSOP_1 theories will be illustrative. Reading: Kim (1998), "Forking in simple unstable theories"; Kim and Pillay (1997), "Simple theores"; Chernikov and Ramsey (2016), "On model-theoretic tree properties"; Kaplan and Ramsey (2020) "On Kim-independence".

The stable forking conjecture: The strongest partial results proving cases of the stable forking conjecture are those of Brower, on types of rank two, and Casanovas and Wagner, on countably categorical CM-trivial theories. The simple Kim-forking conjecture in NSOP_1 theories is amenable to different techniques: an infinite-variable, global variant is true in general, while stronger versions are true under the assumption of finite F_Mb or the definable Morley property, nontrivial examples of which are given by the incidence structures discussed in the next point. Reading: Palacín and Wagner (2013), "Elimination of Hyperimaginaries and Stable Independence in Simple CM-Trivial Theories," Section 4; Brower (2012), "Aspects of stability in simple theories," Sections 2.2 and 2.3; Baldwin, Freitag and Mutchnik (2024), "Simple homogeneous structures and indiscernible sequence invariants," Section 7.

Abstract properties of forking in free constructions and incidence structures: Incidence structures, such as Conant and Kruckman's generic projective planes and Hyttinen and Paolini's free projective planes, later simplified by Baldwin, Mutchnik and Freitag, give nontrivial examples of the definable Morley property and finite F_Mb. The generic projective planes give nontrivial examples of the definable Morley property in an NSOP_1 theory, while the free projective planes give nontrivial examples of 1 < F_Mb(p) < ∞ in a stable theory. The quantity F_Mb is also connected to Baldwin, Freitag and Mutchnik's proof of the Koponen conjecture, and specifically the case of showing that every supersimple theory with quantifier elimination in a finite relational language has trivial forking. Should such a theory have nontrivial forking, Tomašić and Wagner's pseudolinearity theorem would produce a type p for which F_Mb(p) = ∞, getting a contradiction. Proving the pseudolinearity theorem from the group configuration theorem for countably categorical simple theories demonstrates the unexpected relationship between model-theoretic algebra and pure model-theoretic statements about forking.

   In a different direction, Dobrowolski, Kim and Ramsey extend the theory of Kim-independence in NSOP_1 theories from an independence relation over models to an independence relation over arbitrary sets, assuming that the theory satisfies the existence axiom: every type over a set has a forking extension to every larger set. They leave as an open problem whether every NSOP_1 theory satisfies the existence axiom. Mutchnik answers this question, finding an NSOP_1 theory without the existence axiom using a variation on a specific free construction, the free pseudoplane discussed in, say, Pillay (2001), Geometric Stability Theory.

   Readings: Conant and Kruckman (2019), "Independence in generic incidence structures"; Hyttinen and Paolini (2021), "First-order model theory of free projective planes"; assorted readings from Baldwin, Freitag and Mutchnik (2024), "Simple homogeneous structures and indiscernible sequence invariants" including section 4 (an incidence structure with 1 < F_Mb(p) < ∞ in a stable  theory), section 3.2 (giving an exposition of pseudolinearity and connections to F_Mb), the proof of the supersimple and finite rank cases of Theorem 6.2 (applying pseudolinearity and F_Mb to the Koponen conjecture on simple homogeneous structures), Examples 7.22 and 7.34 (on the definable Morley property); Tomašić (2001) "Geometric simplicity theory," section 5.5; Dobrowolski, Kim and Ramsey, "Independence over arbitrary sets in NSOP_1 theories"; Mutchnik (2024), ``An NSOP_1 theory without the existence axiom".

Forking outside of simple and NSOP_1 theories: To make sense of forking in Shelah's NSOP_n hierarchy for n > 1, new generalizations of forking-independence, including Conant-independence and n-ð-independence, are promising. Mutchnik shows that n = 4 is the largest (integer) value of n such that all theories with symmetric Conant-indepndence are NSOP_n, and, proving an analogous result, demonstrates a connection between n-ð-independence and the exponential levels NSOP_{2^{n+1}+1} of Shelah's hierarchy. In another direction, Chernikov and Kaplan prove that forking is equal to dividing in any NTP_2 theory. Readings: Mutchnik (2024), "Conant-independence and generalized free amalgamation", Sections 5 and 6; Mutchnik (2023), "On the properties SOP_{2^{n+1}+1}"; Chernikov and Kaplan (2012), "Forking and dividing in NTP_2 theories".

Understanding the syntactic via the semantic: Chernikov, Kaplan and Ramsey use Kim-independence to show that every binary NSOP_1 theory is simple. Baldwin, Freitag and Mutchnik use forking-independence to prove that every simple theory with quantifier elimination in a finite relational language is supersimple, which has an entirely syntactic statement in terms of trees. Mutchnik uses Conant-independence to show that NSOP_1 is equal to NSOP_2 Readings: Kaplan, Ramsey and Simon (2023), "Generic stability independence and treeless theories"; Baldwin, Freitag and Mutchnik (2024), "Simple homogeneous strctures and indiscernible sequence invariants," Theorem 6.2; Mutchnik (2022), ``On NSOP_2 theories".