The PKU model theory seminar takes place on Mondays from 2-3 either Online or in Room 09 at Quan Zhai, BICMR (全斋9号教室).
Upcoming Talks:
Wednesday, December 11th, 2024
Speaker: TBA
Title: TBA
Abstract: TBA
Wednesday, December 18th, 2024
Speaker: Tomasz Rzepecki (Uniwersytet Wrocławski) (zoom)
Title: TBA
Abstract: TBA
Old Talks:
Wednesday, November 20th, 2024
Speaker: James Hanson (Iowa State) (zoom)
Title: Special coheirs and model-theoretic trees
Abstract: We will discuss a characterization of first-order theories realizing a certain combinatorial tree configuration in terms of special coheirs.
Wednesday, November 13th, 2024
Speaker: Atticus Stonestrom (University of Notre Dame) (zoom)
Title: F-generic types in NIP groups
Abstract: In stable group theory, a key role is played by "generic" types. In the more general setting of definably amenable NIP groups, these no longer suffice, and the appropriate substitute is given by "f-generic" types. I will present a positive answer to a question of Chernikov and Simon, showing that, for an NIP group, definable amenability is equivalent to the existence of f-generic types. As a quick application, I will show any dp-minimal group is definably amenable.
Wednesday, November 6th, 2024
Speaker: Nigel Pynn-Coates (University of Vienna) (zoom)
Title: Tame pairs of transseries fields
Abstract: Transseries emerged in connection with Écalle’s work on Dulac’s problem and Dahn and Göring’s work on nonstandard models of real exponentiation, and some can be viewed as asymptotic expansions of solutions to differential equations. More recently, Aschenbrenner, Van den Dries, and Van der Hoeven completely axiomatized the elementary theory of the differential field of (logarithmic-exponential) transseries and showed that it is model complete. This talk concerns pairs of models of this theory such that one is a tame substructure of the other in a certain sense. I will describe the model theory of such transserial tame pairs, including a model completeness result for them, which can be viewed as a strengthening of the model completeness of large elementary extensions of the differential field of transseries, such as hyperseries, surreal numbers, or maximal Hardy fields.
Wednesday, October 30th, 2024
Speaker: Chuyin Jiang (UCLA) (zoom)
Title: Indiscernible Subsequence in NIP Theories
Abstract: Ramsey's theorem, which establishes the existence of monochromatic cliques in edge-colored complete graphs, forms the foundation of Ramsey theory, a field dedicated to uncovering patterns within seemingly disordered structures. As the scope of the problem expands to include multiple colorings, hypergraphs, and infinite sets, the existence of Ramsey cardinals cannot be established within ZFC. In this presentation, we reinterpret n-types as colorings of ordered sequences, where a "monochromatic clique" corresponds to an indiscernible subsequence. We will discuss the implications of this framework for the existence of Ramsey cardinals in NIP theories.
Wednesday, October 23rd, 2024
Speaker: Paolo Marimon (TU Wien) (zoom)
Title: When invariance implies exchangeability
Abstract: Joint work with Samuel Braunfeld and Colin Jahel. We study invariant random expansions of a structure $M$ by some hereditary class of structures. For example, we study $\mathrm{Aut}(M)$-invariant (regular) Borel probability measures on the space of graphs on the domain of $M$. We prove that for many homogeneous structures of arity $>2$, all invariant random expansions of $M$ by graphs are exchangeable, i. e. invariant under all permutations. The latter are well-understood due to the Aldous-Hoover theorem in probability. More generally, we define a measure of randomness of $M$, $k$-overlap closedness, which implies that invariant random expansions by hereditary classes of sufficiently slow growth rate, i.e. $O(e^{n^{k+\delta}})$ for all $\delta>0$, are exchangeable. Since invariant Keisler measures are a special case of invariant random expansions, our work allows us to describe the spaces of invariant Keisler measures of various homogeneous structures of arity $>2$. In particular, we show there are $2^{\aleph_0}$ supersimple homogeneous ternary structures for which there are non-forking formulas which are universally measure zero.
Wednesday, October 9th, 2024
Speaker: Michal Szachniewicz (University of Oxford) (zoom)
Title: Foundations of globally valued fields.
Abstract: I will present results from a joint work with Itaï Ben Yaacov, Pablo Destic and Ehud Hrushovski, where we lay foundations of globally valued fields and relate them to Arakelov geometry and to Banach lattices. Globally valued fields (abbreviated GVF) are a class of fields with an extra structure, capturing some aspects of the geometry of global fields, based on the product formula. We give a representation theorem, saying that on a countable fields, every GVF structure comes from a proper adelic curve, as defined by Chen and Moriwaki. If time permits, I will talk about some applications from a joint work with Pablo Destic and Nuno Hultberg.
Monday, June 3rd, 2024
Speaker: Nick Ramsey (University of Notre Dame) (zoom)
Title: Higher amalgamation in PAC fields
Abstract: Pseudo algebraically closed (PAC) fields, introduced by Ax in his characterization of pseudo-finite fields, are an important class of examples in both model theory and field arithmetic. A recurrent theme in the study of PAC fields is that their analysis often reduces to an analysis of the absolute Galois group. One of the most significant results along these lines is a theorem of Chatzidakis, which relates the independence theorem in a PAC field to the independence theorem in the inverse system of the absolute Galois group, viewed as a first-order structure. We explain how this connection can be generalized to relate independent n-amalgamation in a PAC field to independent n-amalgamation in the absolute Galois group. We will describe some corollaries for omega-free PAC fields and, more generally, for Frobenius fields.
Monday, May 27th, 2024
Speaker: Slavko Moconja (University of Belgrade) (Zoom)
Title: Weakly o-minimal types
Abstract: A complete type p(x) is weakly o-minimal (wom) if there exists a relatively definable linear order < on its locus p(C) such that every relatively definable subset of the locus is a union of finitely many <-convex parts. In the talk, I will discuss several problems concerning this notion. These include a description of all relatively definable linear orders on p(C), appropriate Monotonicity theorems for functions relatively definable on p(C), and properties of the forking-dependence relation. Joint work with Predrag Tanović.
Monday, May 20th, 2024
Speaker: Tingxiang Zou (University of Bonn) (Zoom)
Title: Non-expansion of polynomial actions
Abstract: Consider a finite set F of one-variable polynomials of bounded degree over the complex numbers. Let X be a finite subset of the complex numbers and F*X be the collection of f(x) for f in F and x in X. Suppose the sizes of F and X are comparable in the sense that |F| is bounded above and below by some fixed powers of |X|. We want to ask when it is possible that |F*X|<|X|^{1+e} for very small e>0? The goal of this talk is to answer this question by considering a generalization of the Elekes-Szabó's Theorem. This is joint work with Martin Bays.
Monday, May 13th, 2024
Speaker: Chieu-Mihn Tran (National University of Singapore)
Title: Growth in groups: direct and inverse problems
Abstract: This largely survey talk is a prequel to the talk “Measure doubling of small sets in SO(3,R)” (joint work with Jing Yifan and Zhang Ruixiang) given in Beijing Logic Meeting 2023. I will explain why the use of model theory is not accidental in such combinatorics flavored results and is to some degree the same reason we can show that large stable fields are separably closed (joint work with Will Johnson, Erik Walksberg, and Ye Jinhe). More mathematically, we consider a group G equipped with a notion of size (cardinality, Haar measure, etc), and let A be a subset of G. The following problems are of interest: (i) Find inequalities relating the size of the product set A^k ={a_1 a_k: a_1, a_2 \in A} and the size of A. (ii) Understand when the size of A^k is not too large compared to that of A. These are called the direct and inverse problems for growth in groups. I will explain how these problems arise independently in many areas including number theory, additive combinatorics, convex geometry, analysis, geometric group theory, and model theory. Beside the result about SO(3,R), I will also discuss some other recent progresses in addressing them including the solution of the Polynomial Freiman-Ruzsa Conjecture for (F_2)^n by Gowers, Green, Manners, Tao and sharp stability results for Brunn—Minkowski theorem by Figalli, van Hintum, and Tiba.
Monday, May 6th, 2024
Speaker: David Meretzky (University of Notre Dame) (Zoom)
Title: Differential Field Arithmetic
Abstract: I will discuss some of my upcoming thesis work under the supervision of Anand Pillay. Some of this work is also joint with Omar León Sánchez. Motivated by existence questions in differential Galois theory, I will discuss our recent efforts to generalize a theorem of Serre from the algebraic to the differential algebraic setting. Serre's theorem states: A field F is bounded (has finitely many extensions of each finite degree) if and only if the first Galois cohomology set with coefficients in a linear algebraic group defined over F is trivial. This talk will emphasize our development of basic computational tools for definable Galois cohomology, a model theoretic generalization of (differential) algebraic Galois cohomology. All of the relevant notions will be introduced including some background on differential Galois theory.
Monday, April 29th, 2024
Speaker: Christian d'Elbée (University of Leeds) (Zoom)
Title: Existentially closed nilpotent Lie algebras
Abstract: I will present ongoing work joint with Müller, Ramsey and Siniora. A classical result of Macintyre and Saracino states that the theory of Lie algebras over a fixed field and of bounded nilpotency class does not admit a model-companion. We prove that by letting the field grow (i.e. with a separated sort for the field) the theory of Lie algebras of bounded nilpotency class admits a model-companion and that this theory relates asymptotically to the omega-categorical existentially closed c-nilpotent Lie algebra over a finite field F_p for c < p. We also prove that if the theory of the field is NSOP1 then the theory of the corresponding Lie algebra is NSOP4. We will explain how to get this result via a criterion for NSOP4 which does not use stationary independence relations.
Monday, April 22nd, 2024
Speaker: Aaron Anderson (UCLA) (zoom)
Title: Generically Stable Measures in Continuous Logic
Abstract: Continuous logic generalizes first-order logic to take real truth values, and Keisler measures similarly generalize types. We combine these two generalizations to study measures in continuous logic, focusing on generically stable measures, which are particularly well-behaved. In NIP theories, generically stable measures have several equivalent definitions. We translate these definitions to continuous logic, and show that they remain equivalent in this context. This requires developing real-valued versions of combinatorial theorems about classes of finite VC-dimension, and extrapolating definitions like smoothness and weak orthogonality of measures to continuous logic. We then use generically stable measures to derive regularity lemmas for NIP and distal metric structures.
Monday, April 15th, 2024
Speaker: Zhentao Zhang (Fudan University)
Title: Newelski's question on definably amenable groups over p-adics
Abstract: Let G be a definably amenable groups over the p-adics. We study the definable topological dynamics of (G,S_{G}(\mathbb{Q}_{p})) of G acting on its type space. We focus on the question of whether weakly generic types coincide with almost periodic types which was raised by Newelski and then restarted by Chernikov and Simon on NIP definably amenable groups. We show that the stationarity is a sufficient and necessary condition for the positive answer of the question. This is a joint work with Ningyuan Yao.
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.