PKU model theory seminar

The PKU model theory seminar takes place on Mondays from 2-3 in Room 09 at Quan Zhai, BICMR (全斋9号教室).

Upcoming Talks:

Monday, April 15th, 2024

Speaker: Zhentao Zhang (Fudan University)

Title: TBA

Abstract: TBA

Monday, April 22nd, 2024

Speaker: Aaron Anderson (UCLA) (zoom)

Title: TBA

Abstract: TBA

Monday, April 29th, 2024

Speaker: Christian d'Elbée (University of Leeds) (Zoom)

Title: TBA

Abstract: TBA

Old Talks:

Monday, April 8th, 2024

Speaker: Patrick Lutz (UC Berkley) (Zoom)

Title: A theory which *really* doesn't have a computable model

Abstract: The completeness theorem guarantees that any consistent theory has a model, but it is not hard to show that a computable version of this statement does not hold: there is a computable, consistent theory with no computable model. For example, a slight modification of the proof of Tennenbaum's Theorem implies that the theory ZF has no computable model. Recently, however, Pakhomov discovered an interesting limitation of this example: there is a theory which is definitionally equivalent to ZF and has a computable model ("definitional equivalence" is a strong form of bi-interpretability). One can interpret this result as saying that if we just formulate ZF in a different (but equivalent) language, then it does have a computable model. In light of this, Pakhomov raised the question of whether every computable consistent theory is definitionally equivalent to a theory with a computable model. James Walsh and I have constructed a counterexample to Pakhomov's question---a computable, consistent theory for which no definitionally equivalent theory can have a computable model. The proof uses a mix of computability theory and tame model theory, in the guise of Laskowski's theory of mutual algebraicity. I will discuss the background of our result and some of the ingredients of the proof, focusing on the role played by model theory.

Monday, April 1st, 2024

Speaker: Hu Yuqi (Tsinghua University)

Title: Model theoretic version of Szemerédi's regularity lemma

Abstract: Szemerédi's regularity lemma is a fundamental result in graph combinatorics. But the bound can be unavoidably large for general (hyper)graphs. On the other hand, the result can be viewed as results about hypergraphs with the edge relation definable. When its first-order theory satisfies some tameness condition in model theory, we can get better bound for the result. This is an expository talk on the paper `the definable regularity lemmas for NIP hypergraphs' by Artem Chernikov and Sergei Starchenko.

Monday, March 25th, 2024

Speaker: Shichang Song (Beijing Jiaotong University)

Title: Model theory of random variables

Abstract: The class of [0,1]-valued random variables on atomless probability spaces is an elementary class in continuous logic. The theory of this class is denoted by ARV. ARV is complete, separably categorical, omega-stable, and admits quantifier elimination. During this talk, we will characterize saturated models of ARV, and give explicit formulas between types. Finally, we will discuss type spaces of ARV and Wasserstein spaces.

Monday, March 18th, 2024

Speaker: Rizos Sklinos (AMSS, CAS)

Title: First-order sentences in random groups

Abstract: Gromov in his seminal paper introducing hyperbolic groups claimed that a “typical” finitely presented group is hyperbolic. His statement can be made rigorous in various natural ways. The model of randomness that is preferentially focused on is Gromov's density model, as it allows a fair amount of flexibility. In this model a random group is hyperbolic with overwhelming probability. In a different line of thought, Tarski asked whether all non abelian free groups share the same first-order theory (in the language of groups). This question proved very hard to tackle and only after more than 50 years Sela and Kharlampovich-Myasnikov answered the question positively. Combining the two, J. Knight conjectured that a first-order sentence holds with overwhelming probability in a random group if and only if it is true in a no abelian free group. In joint work with O. Kharlampovich we answer the question positively for universal-existential sentences.

Monday, March 11th, 2024

Speaker: Wei Li (AMSS, CAS)

Title: Effective Definability of Kolchin Polynomials

Abstract: While the natural model-theoretic ranks available in differentially closed fields of characteristic zero, namely Lascar and Morley rank, are known not to be definable in families of differential varieties; in this talk we show that the differential-algebraic rank given by the Kolchin polynomial is in fact definable. This result relies on a uniform bound on the Hilbert-Kolchin index. As a byproduct, we show that the property of being weakly irreducible for a differential variety is also definable in families. The question of full irreducibility remains open; it is known to be equivalent to the generalized Ritt problem.

Monday, March 4th, 2024

Speaker: Kyle Gannon (BICMR)

Title: Model theoretic events

Abstract: This talk is motivated by the following two soft questions: How do we sample an infinite sequence from a first order structure? What model theoretic properties might hold on almost all sampled sequences? We advance a plausible framework in an attempt to answer these kinds of questions. The central object of this talk is a proability space. The underlying set of our space is a standard model theoretic object, namely the space of types in countably many variables over a monster model. Our probability measure is an iterated Morley product of a fixed Borel-definable Keisler measure. Choosing a point randomly in this space with respect to our distribution yields a random generic type in infinitely many variables. We are interested in which model theoretic events hold for almost all random generic types. Two different kinds of events will be discussed: (1) The event that the induced structure on a random generic type is isomorphic to a fixed structure; (2) the event that a random generic type witnesses a dividing line.