Abstracts

Chatzidakis, Zoé (CNRS(IMJ-PRG), Université Paris Cité)

Title: Old and new results about PAC fields

Abstract: The model theory of PAC fields is important, both in the study of fields (they are the only non-classical fields with potentially a decidable theory), and in pure model theory (they provided the first examples of unstable fields with a simple theory, and of non-simple fields with an NSOP1 theory). The concept of PAC structures was also generalised to other classes of structures than fields. The talk will be a survey of known results, with an emphasis on results with a pure model theoretic flavour.

Chernikov, Artem (University of California, Los Angeles)

Title: Regularity lemma for slice-wise stable hypergraphs

Abstract: We discuss various strengthenings of Szemerédi's regularity lemma for hypergraphs that are tame from the model-theoretic point of view. Generalizing the case of stable graphs due to Malliaris and Shelah, we have shown the following in a joint work with Starchenko: if a 3-hypergraph E(x,y,z) on X × Y × Z is stable when viewed as a binary relation under any partition of its variables in two groups, then there are partitions Xi of X, Yj or Y and Zk of Z so that the density of E on any box Xi × Yj × Zk is either 0 or 1. Terry and Wolf raised the question if the assumption can be relaxed to slice-wise stability, i.e. for any z in Z, the corresponding fiber Ez is a stable relation on X × Y, and similarly for any permutation of the variables (analogous slice-wise assumption is known to be correct in the NIP case). We provide an example of a slice-wise stable 3-hypergraph which does not satisfy the stable regularity lemma above, and establish an optimal weaker partition result for slice-wise stable hypergraphs. Joint work with Henry Towsner.

de la Nuez Gonzalez, Javier (Korea Institute for Advanced Study)

Title: Some model theory of the curve graph

Abstract: The curve graph of a surface of finite type is a fundamental object in the study of its mapping class group both from the metric and the combinatorial point of view. I will discuss joint work with Valentina Disarlo and Thomas Koberda where we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis on the problem of interpretability between different curve graphs and other geometric complexes.

Goodrick, John (Universidad de los Andes)

Title: Type amalgamation properties and homology groups: a retrospective and future directions

Abstract: In theories with a well-behaved independence relation, various kinds of type amalgamation properties have been studied: most famously, the Independence Theorem for nonforking in simple theories (or Kim independence in NSOP1 theories), which is also known as "3-existence" since it can be seen as the existence of an amalgamation problem for types of three independent elements. Generalizing this idea, we obtained some results on higher-dimensional type amalgamation properties in simple theories such as n-existence and n-uniqueness in joint work with Byunghan Kim and Alexei Kolesnikov a decade or so ago, also developing a notion of homology groups which measure failures of amalgamation.

We will revisit these results in the light of recent developments in the study of other independence relations and propose some avenues for future work.

Kamsma, Mark (Queen Mary University of London)

Title: Dividing lines between positive theories

Abstract: This is joint work with Francesco Gallinaro and Anna Dmitrieva. We give definitions of the properties OP, IP, k-TP, TP1, k-TP2, SOP1, SOP2 and SOP3 in positive logic, a proper generalisation of full first-order logic where negation is not built in, but can be added as desired. We prove various implications and equivalences between these properties, matching those known from full first-order logic. We will look at an example illustrating why these definitions have to be altered for positive logic. We will also give positive versions of the well-known results that a theory is stable iff it is NSOP and NTP2, and that a theory is simple iff it is NTP1 and NTP2. In the positive versions of these theorems we replace NSOP and NTP1 respectively by NSOP1. This allows us to give completely new proofs based on Kim-independence.

Kaplan, Itay (Hebrew University of Jerusalem)

Title: NSOP3 = simple for binary theories

Abstract: (Joint work with Pierre Simon and Nick Ramsey) I will present the idea of a proof that when the theory is binary or close to being binary then NSOP3 and simplicity collapses.

Kolesnikov, Alexei (Towson University)

Title: Private machine learning and model theory

Abstract: Let X be a set and C be a family of subsets of X. One of the ways to formalize a learning algorithm is as follows. A learning algorithm inputs a finite set, called sample, of points of X together with information whether they belong to some c*∈C; the output is an element of C or a probability distribution over C. To be of practical use, the output of the learning algorithm needs to be reasonably accurate, for example, be probably approximately correct (PAC); the output should not change significantly if a single element in a sample changes (a differential privacy condition), and the size of the sample cannot get too large.

A recent advance by Alon, Bun, Livni, Malliaris, and Moran established that a family of sets can be PAC-learned with differential privacy if and only if the family of sets has a finite Littlestone dimension, a property closely connected to model-theoretic stability. I will discuss a recent project with Vince Guingona and several REU students, Avery Schweitzer, Julie Nierwinski, and Richard Soucy, that explored different notions of differential privacy and bounds on the learning algorithm sample size as a function of the Littlestone dimension.

Krupiński, Krzysztof (Uniwersytet Wrocławski)

Title: Ramsey theory and topological dynamics for first order theories, and an abstract generalization

Abstract: In the first part of the talk, I will discuss the theory developed in my joint paper [1] with Junguk Lee and Slavko Moconja which can be viewed as a variant of the celebrated Kechris, Pestov, and Todorčević theory (shortly, KPT theory) [2] in the context of (complete first order) theories. This leads to correspondences between Ramsey-theoretic properties which involve “definable colorings” and dynamical properties of the underlying theory, i.e. properties that are expressed in terms of the action of the group of automorphisms of a monster (i.e. sufficiently saturated and homogeneous) model of the theory in question on the appropriate space of types. On the one hand, this leads to counterparts of some results from KPT theory, but on the other hand, to essentially new results. One of the main contributions are combinatorial criteria for triviality and profiniteness of the Ellis group of the theory in question.

In the second part, I will focus on my ongoing project with Junguk Lee and Slavko Moconja, in which we have adapted the definitions of various definable Ramsey properties from [1] to the general context of 0-dimensional ambits (and some of them even to arbitrary ambits) and we have adapted the proofs of the main results of [1] to this general context, obtaining correspondences between Ramsey theoretic properties (with suitable “definability” requirements on colorings) and dynamical properties of the ambit in question and yielding criteria for triviality and profiniteness of the Ellis group of this ambit.

This general abstract context can be applied to various interesting situations, in particular to:

(1) first order theories,
(2) definable groups,
(3) classical KPT theory.

In (1), one recovers the main results from [KLM]. In (2), it gives us new results and leads to some fundamental questions. A very interesting situation arises in (3) with potential new structural results in this classical context and some natural questions.

[1] K. Krupiński, J. Lee, S. Moconja, Ramsey theory and topological dynamics for first order theories, Trans. Amer. Math. Soc. 375 (2022), 2553-2596.
[2] A.S. Kechris, V. G. Pestov, S. Todorčević, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 (2005), 106-189.

Lee, Junguk (Changwon National University)

Title: Preservation of non-antichain tree property

Abstract: The antichain tree property (in short, ATP), which implies both SOP1 and TP2, was introduced by Jinhoo Ahn and Joonhee Kim in their study of relationship between SOP1 and SOP2. In this talk, first, we will see ATP satisfies some useful properties enjoyed by previous dividing lines, for example, witness in one-variable and equivalence of k-ATP and ATP. Based on these properties, we will give some criteria for a first order theory to have ATP or non-ATP (in short, NATP). Second, we will give several examples having NATP and having SOP and TP2 and so they do not fit into previous dividing lines. For example, a Hahn field of a Frobenius field of characteristic 0, the random parametrization of DLO, ACFO, and so on. This talk is based on joint work with Jinhoo Ahn, Joonhee Kim, and Hyoyoon Lee.

Malliaris, Maryanthe (University of Chicago)

Title: Some theorems and open questions in simple theories

Abstract: In honor of the occasion, this talk will discuss some recent theorems and open problems in simple theories which may not be widely known. 

Mutchnik, Scott (University of Illinois Chicago)

Title: (N)SOP2^(n+1)+1 Theories

Abstract: Indiscernible sequences, particularly Morley sequences relative to some definition of independence, contain information about genericity in a first-order theory. However, there is a fine structure to this genericity that is submerged by the existential viewpoint of forking "at a generic scale," as demonstrated by work of Kaplan and Ramsey on witnesses to Kim-forking in NSOP1 theories. We discuss how further approximations of strict order can see this fine structure, which is obtained by iteration, and reveals exponential behavior within Shelah’s NSOPn hierarchy. While recent research has focused on the properties SOP1, SOP2, SOP3, and SOP4 and their negations, little was previously known about SOPn for n greater than 4, other than some examples which distinguish these properties. Our results suggest a potential theory of independence for NSOPn theories, for arbitrarily large values of n.

Pillay, Anand (University of Notre Dame)

Title: Forking and invariant measures

Abstract: This is joint with Atticus Stonestrom. We look at the question of whether a formula which does not fork over 0 has positive measure under some global automorphism invariant Keisler measure, in the context of NIP theories T. We give a positive result, assuming that T is first order amenable, and a counterexample in general. 

Scanlon, Thomas (University of California, Berkeley)

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Scow, Lynn (California State University, San Bernardino)

Title: Semi-retractions, generalized indiscernible sequences, and trees

Abstract: In my joint paper with Byunghan Kim and Hyeung-Joon Kim ``Tree indiscernibilities, revisited" we explicated mechanisms for deducing the Ramsey property for a class K2 from a class K1 of trees known to have the Ramsey property.  The mechanisms for these transfers of the Ramsey property became the focus of several of my subsequent papers.  In particular, I defined the notion of a "semi-retraction'' to capture some of the qualities of this transfer in a way that was tied to techniques around generalized indiscernible sequences.  In recent work, Dana Bartošová and I have found a universal algebraic proof for how semi-retractions transfer Ramsey-type properties in our preprint "A new perspective on semi-retractions and the Ramsey property.''  In this talk, I plan to outline some of these developments.

Tsuboi, Akito (University of Tsukuba)

Title: Expanding Random Structures: Colorings and Automorphisms

Abstract: In this talk, we discuss random structures, which are the Fraisse limits of classes of finite structures. A classic example of such structures is the countable random graph. We are interested in two types of expansions of these random structures.

Firstly, we examine expansions using predicate symbols for coloring. Specifically, we focus on countable random graphs that are edge-colored with a finite number of colors. Among other results, we demonstrate that if these graphs do not contain a monochromatic random subgraph, there exists a random subgraph that exhibits the strict order property in the expanded language. Furthermore, in such cases, there are infinitely many independent strict orders.

Regarding expansions by an automorphism, we particularly concentrate on the presence of a cyclic automorphism. Some of the results mentioned are obtained through joint work with H. Kikyo. It is worth noting that many countable random structures have a cyclic automorphism, while structures constructed using Hrushovski's construction lack such an automorphism.

Wagner, Frank (Université Claude Bernard Lyon 1)

Title: The model theory of skew braces

Abstract: Skew braces were introduced as algebraic structures encoding set-theoretic solutions to the Young-Baxter equation from physics. They are structures whose domain B supports two (non-commutative) group operations + and ·, with a left pseudo-distributive law a·(b+c)=a·b-a+a·c. Their algebraic properties are developed in analogy with the theory of rings, but additional problems arise since the distributive law is unilateral, and not an additive homomorphism. I shall introduce the basic properties of skew braces, and present some results on stable, ω-stable and ω-categorical skew braces. (Joint work with Maria Ferrara, Marco Trombetti and Moreno Invitti)

Zou, Tingxiang (University of Münster)

Title: Number of rational points of difference varieties in finite difference fields

Abstract: A difference field is a field with a distinguished automorphism. An automorphism of a finite field is a power of the frobenius map. In this talk, I will discuss how to estimate the number of rational points of a difference variety, namely the size of the solution set of a system of difference equations, in a finite field with a distinguished power of frobenius. Like algebraic geometry, one can assign a dimension, called transformal dimension, to a difference variety. I will present a result which is a difference version of the Lang-Weil estimate with respect to the transformal dimension. This is joint work with Martin Hils, Ehud Hrushovski and Jinhe Ye.

Poster Session