Schedule for: 18w5193 - Neostability Theory

Let G be a real algebraic unipotent group and let Lambda be a lattice in G, with p:G->G/Lambda the quotient map. Given a definable subset X of G, in some o-minimal expansion of the reals, we describe the closure of p(X) in G/Lambda in terms definable families of cosets of real algebraic subgroups of G of positive dimension. The family is extracted from X independently of Lambda.

Adler, Casanovas and Pillay proved that if p is a complete stable type over a set B which does not fork over a set A, then the restriction of p to A is also stable. I will address the analogous question, replacing stable with NIP. In addition I will present a new proof for the stable case which uses elementary techniques.

In recent work with Krupiński, we showed that strong type spaces can be seen (in a strong sense) as quotients of compact Polish groups, and as a consequence. I will give a brief account of the argument, as well as describe some applications, such as showing that a non-definable analytic subgroup of a type-definable group has index continuum (and in particular, that an analytic subgroup cannot have countably infinite index).

In their work on the model theory of algebraically closed valued fields, Haskell, Hrushovski and Macpherson developed a notion of stable domination and metastability which tries to capture the idea that in an algebraically closed valued field, numerous behaviors are (generically) controlled by the value group and/or the residue field. In this talk I will explain how (finite rank) metastability can be used to decompose commutative definable groups, in term of stable groups and value group internal groups. Time permitting, I will quickly describe the applications of these results to the study of algebraically closed valued fields, in particular, the classification of interpretable fields.

I will introduce a framework that generalises usual zariski geometries, quantum zariski geometries as well as other non algebraic examples and non-reduced ringed spaces generalising algebraic varieties, compact complex manifolds and non Archimedean geometry.

By a result of Simon it is known that a theory of a coloured linear order has quantifier elimination after naming all unary $L$-definable sets and all $L$-definable binary monotone relations. Motivated by this result we define the notion of monotone theories, theories of linear orders in which binary definable sets have previous description. More precisely, an $\aleph_0$-saturated structure $M$ is said to be monotone if there exists an $L$-definable linear order $<$ such that every $A$-definable subset of $M^2$ is a finite Boolean combination of unary $A$-definable sets and $A$-definable $<$-monotone relations, in which case we also say that $M$ is monotone with respect to $<$. A theory is said to be monotone if it has a monotone $\aleph_0$-saturated model. We prove that the class of monotone theories coincides with the class of weakly quasi-o-minimal theories introduced by Kudaibergenov. Moreover, we describe definable linear orders in monotone theories and show that monotonicity of a theory does not depend on the choice of the linear order. Joint work with Predrag Tanovi\'c.

A longstanding open questions asks whether an unstable NIP theory interprets an infinite linear order. I will present a construction giving a type-definable linear (quasi-)order, thus partially answering this question.

I will discuss some aspects of my recent paper (still in preparation) with Udi Hrushovski and Anand Pillay. Most of the main results are of the form "a version of amenability implies a version of G-compactness". Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some extension results that we obtain for measure-like functions (which we call means and pre-means). One of the main results is the following. Assume $G$ is a 0-definable group normalized by $G(M)$ (where $M$ is a model), $H$ is a 0-type-definable subgroup, and $N$ is the normal subgroup of $G$ generated by $H$. Then, if the type space $S_{G/H}(M)$ carries a $G(M)$-invariant, Borel probability measure, then $G^{00}_M \leq NG^{000}_M$. We also obtain a similar result which answers various questions from my earlier paper with Anand Pillay, e.g. if G is an amenable topological group, then the (classical) Bohr compactification of $G$ coincides with a certain "weak Bohr compactification". In other words, the conclusion says that certain connected components of $G$ coincide: $G^{00}_{top} = G^{000}_{top}$. Another main result says that each amenable theory is G-compact. In particular, we introduce the notion of an amenable theory in several equivalent ways, e.g. by saying that for each type $p(\bar x) \in S(\emptyset)$, the space $S_p(\C):={q \in S(\C): p \subset q}$ carries an $Aut(\C)$-invariant, Borel probability measure (where $\C$ is a monster model). There are several other developments in this paper, but during my talk, I will focus on one of the above.

The notion of the localized Lascar-Galois group $Gal_L(p)$ of a type $p$ appeared recently in the context of model-theoretic homology groups, and was also used by Krupinski, Newelski, and Simon in the context of topological dynamics. After a brief introduction of the context, we will discuss some basic properties of localized Lascar-Galois groups. Then, we will focus on the question about how far $Gal_L(tp(acl(a)))$ can be from $Gal_L(tp(a))$. This is a joint work with B. Kim, A. Kolesnikov and J. Lee.

Fix languages L_1 and L_2 with intersection L_\cap and union L_\cup. An L_\cup structure M is interpolative if whenever X_1 is an L_1-definable set and X_2 is an L_2-definable set, X_1 and X_2 intersect in M unless they are separated by L_\cap-definable sets. When T_1 is an L_1 theory and T_2 is an L_2 theory, we say that a theory T_\cup^* is the interpolative fusion of T_1 and T_2 if it axiomatizes the class of interpolative models of the union theory T_\cup. If T_1 and T_2 are model-complete, this is exactly the model companion of T_\cup. Interpolative fusions provide a unified framework for studying many examples of "generic constructions" in model theory. Some, like structures with generic predicates, or algebraically closed fields with several independent valuations, are explicitly interpolative fusions, while others, like structures with generic automorphisms (e.g. ACFA), or fields with generic operators (e.g. DCF), are bi-interpretable with interpolative fusions. In joint work with Erik Walsberg and Minh Tran, we study two basic questions: (1) When does the interpolative fusion exist, and how can we axiomatize it? (2) How can we understand properties of the interpolative fusion T_\cup^* in terms of properties of the theories T_1, T_2, and T_\cap? In this talk, I will focus on the latter question. Under mild stability-theoretic assumptions on the base theory T_\cap, we show preservation of a weak form of quantifier-elimination. And using this, we show that the interpolative fusion of NSOP_1 theories is NSOP_1. I will also discuss sufficient conditions for the preservation of other properties of interest (e.g. stability, NIP, simplicity, and NTP_2).

Let $T$ be an NSOP$_1$ theory. Recently I. Kaplan and N. Ramsey proved that in $T$, the so-called Kim-independence ($\phi(x,a_0)$ Kim-divides over $A$ if there is a Morley sequence $a_i$ such that $\{\phi(x,a_i)\}_i$ is inconsistent) satisfies nice properties over models such as extension, symmetry, and type-amalgamation. In a joint work with J. Dobrowolski and N. Ramey we continue to show that in $T$ with nonforking existence, Kim-independence also satisfies the properties over any sets, in particular, Kim’s lemma, and 3-amalgamation for Lascar types hold. Modeling theorem for trees in a joint paper with H. Kim and L. Scow plays a key role in showing Kim’s lemma. If time permits I will talk about a result extending the non-finiteness (except 1) of the number of countable models of supersimple theories to the NSOP$_1$ theory context.

The theory of an algebraically closed field of fixed positive characteristic, expanded by a predicate for an additive subgroup, admits a model companion, which we call ACFG. In this talk, we investigate this example of a strictly NSOP1 theory, and describe some features such as forking and Kim-forking. This construction can be generalised to provide further interesting examples of NSOP1 theories, for example the theory of an algebraically closed field with a generic multiplicative subgroup.

There are multiple connections between model-theoretic notions of complexity and machine learning. NIP formulas correspond to PAC-learning by way of VC-dimension, and stable formulas correspond to online learning by way of Littlestone dimension, also known as Shelah's 2-rank. We explore a similar connection between formulas without the finite cover property and equivalence query learning. In equivalence query learning, a learner attempts to identify a certain set from a set system by making hypotheses and receiving counterexamples. We use the notion of (strong) consistency dimension, an analogue of the the negation of the finite cover property for set systems. We show that finite (strong) consistency dimension and finite LIttlestone dimension characterize equivalence query learning, drawing on ideas from model theory. We also discuss the role of Littlestone dimension and strong consistency dimension in algorithms.

We give (equivalent) friendlier definitions of classifiable theories strengthen known results about how an independent triple of models can be completed to a model. As well, we characterize when the isomorphism type of a weight one extension $N/M$ is uniquely determined by the non-orthogonality class of the relevant regular type and discuss when $N$ is prime over $Ma$ for some finite $a\in N$. This is part of an ongoing project with Elisabeth Bouscaren, Bradd Hart, and Udi Hrushovski.

It is by now almost folklore that if T is a countably categorical theory, and M its unique countable model, then the topological group G(T) = Aut(M) is a complete invariant for the bi-interpretability class of T . This gained renewed interest recently, given the correspondences between dynamical properties of G(T) and classification-theoretic properties of T . From a model-theoretic point of view, the obvious drawback is the restriction to countably categorical theories. As a first step, I will discuss how to generalise the original result to arbitrary theories in a countable language.

Following the important work done on pseudofinite fields and groups of deducing algebraic properties from model theoretic ones we will present some remarks about algebraic properties of pseudofinite rings taht can be obtained from model theoretic considerations. When we use the term pseudofinite ring, group or field we understand an ultraproduct of finite rings, groups or fields respectively.

This is a joint work with Alf Onshuus and Pierre Simon. In this talk we focus on groups with f-generic types definable in NTP2 theories. In particular we study the case of bounded PRC fields. PRC fields were introduced by Prestel and Basarav as a generalization of real closed fields and pseudo algebraically closed fields, where we admit having several orders. We know that the complete theory of a bounded PRC field is NTP2 and we have a good description of forking. We use some alternative versions of Hrushovski’s “Stabilizer Theorem” to describe the definable groups with f generics in PRC fields. The main theorem is that such a group is isogeneous with a finite index subgroup of a quantifier-free definable groups. In fact, the latter group admits a definable covering by multi-cells on which the group operation is algebraic. This generalizes similar results proved by Hrushovski and Pillay for (not necessarily f-generic) groups definable in both pseudo finite fields and real closed fields.

A field K is called pseudo-algebraically closed (PAC) if every absolutely irreducible variety defined over K has a K-rational point. These fields were introduced by Ax in his characterization of pseudo-finite fields and have since become an important object of model-theoretic study. A remarkable theorem of Chatzidakis proves that, in a precise sense, independent amalgamation in a PAC field is controlled by independent amalgamation in the absolute Galois group. We will describe how this theorem and a graph-coding construction of Cherlin, van den Dries, and Macintyre may be combined to construct PAC fields with prescribed model-theoretic properties.

We prove that if D=(G,+,\dots) is a strongly minimal non-locally modular group interpretable in an o-minimal expansion of a field and dim(G)=2 then D interprets an algebraically closed field K and D (as a structure) an algebraic group over K with all the induced K-structure. I will discuss some key aspects of the proof that may be of interest on their own right. Joint work with Y. Peterzile and P. Eleftheriou.

I will report on joint work with Pillay and Terry on arithmetic regularity (a group theoretic analogue of Szemeredi regularity for graphs) for sets of bounded VC-dimension in finite groups, which is proved using a local version generic compact domination for NIP formulas in pseudofinite groups. I will then present more recent work on nonabelian versions of certain "inverse theorems" from additive combinatorics, which are proved using pseudofinite model theory, and can be used to give alternate proofs of NIP arithmetic regularity for certain classes of finite groups.

We prove that in a stable theory, some 2-analysable types give rise to type definable groupoids, with some simplicial data attached to them, extending a well-know result linking groups to internal types. We then investigate how properties of these groupoids relate to properties of types. In particular, we expose some internality criteria.

In this talk we introduce a weaker form of bi-interpretability and see how it can be used to transfer the Ramsey property across classes in different first-order languages. This is a special case of a more general theorem about what we will call color-homogenizing embeddings.

With Gianluca Paolini (in preparation), we constructed families of strongly minimal Steiner $( systems for every $k 3$. A quasigroup is a structure with a binary operation such that for each equation $xy=z$ the values of two of the variables determines a unique value for the third. Here we show that the $2^{ Steiner $(2,3)$-systems are definably coordinatized by strongly minimal Steiner quasigroups and the Steiner $(2,4)$-systems are definably coordinatized by strongly minimal $SQS$-Skeins. Further the Steiner $(2,4)$-systems admit Stein quasigroups but depending on the choice of theory may or may not admit a definable binary function and be definably coordinatized by an $ Stein quasigroup. We exhibit strongly minimal uniform Steiner triple systems (with respect to the associated graphs $G(a,b)$ (Cameron and Webb) with varying numbers of finite cycles. We show how to vary the theory to obtain $2$ or $3$-transitivity. This work inaugurates a program of differentiating the many strongly minimal sets, whose geometries of algebraically closed sets may be (locally) isomorphic to the original Hrushovski example, but with varying properties in the object language. In particular, can one organize these geometries by studying the associated algebra. This work differs from traditional work in the infinite combinatorics of Steiner systems by considering the relationship among different models of the same first order theory.

They are virtually finite-by-abelian.