We discuss the relationship between Lie derivatives and the linear differential equations on cotangent spaces of algebraic D-varieties at sharp points. We also take the liberty to give an account of Ax's theorem (which may be useful as an entry point to the subject for students).

The outline of a “Galois theory” for bimeromorphic geometry is here developed, via the study of model-theoretic definable binding groups in the theory CCM of compact complex spaces. As an application, a structure theorem about principal meromorphic bundles with algebraic structure group, and admitting no horizontal subvarieties, is deduced. Examples of algebraic groups arising as binding groups are provided, as is a characterisation of when they are linear. Using binding groups in CCM it is shown that, in contrast to the situation in differentially closed fields, there are many algebraic groups which admit nontrivial definable torsors over acl-closed sets in the theory DCCM of existentially closed differential CCM-structures. A self-contained exposition of the binding group theorem in totally transcendental theories, that emphasises the bitorsorial nature of the construction, is also included.

We make some elementary observations about relative categoricity and the Gaifman property. T will be a complete theory in a countable language L with a distinguished unary predicate P. We will assume L is relational and T has quantifier elimination. For M a model of of T, M^P is the substructure of M with universe P(M), and T^P is the common L-theory of these M^P. T is said to be relatively categorical if for any models M_1, M_2 of T any isomorphism between M_1^P and M_2^P lifts to an isomorphism between M_1 and M_2. T has the Gaifman property (or P-existence) if every model of T^P is of the form M^P for a model M of T. It was conjectured that if T is relatively categorical then T has the Gaifman property. T is said to be relatively (omega, omega) categorical if relative categoricity holds when restricted to countable models of T. We observe that (i) if T is relatively (omega, omega) categorical then any model of T^P of cardinality at most aleph_1 is of the form M^P for M a model of T, and (ii) if in addition every model M of T is in the algebraic closure of P(M) together with a (finite) subset of M, then T is relatively categorical and has the Gaifman property.

We study groups definable in existentially closed geometric fields with commuting derivations. Our main result is that such a group can be definably embedded in a group interpretable in the underlying geometric field. Compared to earlier work of the first two authors toguether with K. Peterzil, the novelty is that we also deal with infinite dimensional groups.

The main result of the paper extends the well-known Galois correspondence between closed subgroups of Aut(acl^eq(A)/A) and dcl-closed intermediate sets A ⊆ C ⊆ acl^eq(A) to the setting of the minimal closure inside of a prime model of a totally transcendental theory T = T^eq, A ⊆ mcl^eq(A) ⊆ M_A (namely, mcl^eq(A) is the intersection of all self elementary embeddings of the prime model M_A). Precisely, for B a normal intermediate extension A ⊆ B ⊆ mcl^eq(A), definable subgroups of Aut(B/A) are in Galois correspondence with dcl-closed intermediate sets A ⊆ C ⊆ B. This applies to the “Picard-Vessiot closure" K^PV_∞ (or K_∞) of a differential field K of characteristic 0 with algebraically closed field of constants C_K. We also show that normal differential subfields of K^PV_∞ containing K are “iterated PV-extensions" of K, and the Galois correspondence result above holds for these extensions. This fills in some missing parts of Magid's paper [5]. We also discuss exact sequences 1 → N → G → H → 1, where G = Aut(K_2/K), N = Aut(K_2/K_1) and H = Aut(K_1/K), K_1 is a (maybe infinite type) PV extension of K. K_2 is a (maybe infinite type) PV extension of K_1 and K_2 is normal over K (in the differential closure of K) and again C_K is algebraically closed. Both N and H have the structure of proalgebraic groups over C_K. We show that conjugation by any given element of G is a proalgebraic automorphism of N. Moreover if G splits as a semidirect product N⋊ H, then left multiplication by any fixed element of G is a morphism of proalgebraic varieties N× H → N× H. This improves and extends observations in Section 4 of [5] which dealt with one particular example.

We introduce the notion of first order [extreme] amenability, as a property of a first order theory $T$: every complete type over $\emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure [type] in the same variables. [Extreme] amenability of $T$ will follow from [extreme] amenability of the (topological) group $Aut(M)$ for all sufficiently large $\aleph_{0}$-homogeneous countable models $M$ of $T$ (assuming $T$ to be countable), but is radically less restrictive. First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [5], we prove that if $T$ is amenable, then $T$ is G-compact, namely Lascar strong types and Kim-Pillay strong types over $\emptyset$ coincide. This extends and essentially generalizes a similar result proved via different methods for $\omega$-categorical theories in [14]. In the special case when amenability is witnessed by $\emptyset$-definable global Keisler measures (which is for example the case for amenable $\omega$-categorical theories), we also give a different proof, based on stability in continuous logic.

We give a proof of the existence of generalized definable locally compact models for arbitrary approximate subgroups via an application of topological dynamics in model theory. Our construction is simpler and shorter than the original one obtained by Hrushovski in ``Beyond the Lascar group'', and it uses only basic model theory (mostly spaces of types and realizations of types). The main tools are Ellis groups from topological dynamics considered for suitable spaces of types. However, we need to redevelop some basic theory of topological dynamics for suitable ``locally compact flows'' in place of (compact) flows. We also prove that the generalized definable locally compact model which we constructed is universal in an appropriate category. We note that the main result yields structural information on definable generic subsets of definable groups, with a more precise structural result for generics in the universal cover of $\textrm{SL}_2(\mathbb{R})$.

Let T be a countable complete theory with a distinguished unary predicate P, and let T^P be the theory of the P-parts of models of T with the induced structure. T is said to be relatively categorical or categorical over P if any isomorphism between the P-parts of two models of T lifts to an isomorphism of the models in question. We study the special case of relative categoricity where T is internal to T^P (that is, every model M of T is in the definable closure of P(M) together with additional parameters from M). We first give a structure theory for such T: after passing to T^eq and naming a parameter, T is the same thing as a "pure torsor cover" of T^P, namely simply adjoining to T^P a new sort for a torsor S for a ∅-definable group G in T^P, with no additional structure. We discuss relative stability, or stability over P, and give a characterization of relative stability, superstability, and ω-stability of T in terms of H having the stable chain condition, superstable chain condition, and ω-stable chain condition, respectively.

We give an example of an NIP theory $T$ in which there is a formula that does not fork over $\varnothing$ but has measure $0$ under any global $\varnothing$-invariant Keisler measure, and we show that this cannot occur if $T$ is also first-order amenable.

This paper is about the dfg/fsg decomposition for groups G definable in p-adically closed fields. It is proved that for G definably amenable, G has a definable normal dfg subgroup H such that the quotient G/H is a definable fsg group. The result was known for groups definable in o-minimal expansions of real closed fields (see ). We also give a version for arbitrary (not necessarily definably amenable) groups G definable in p-adically closed fields: there is a definable dfg subgroup H of G such that the homogeneous space G/H is definable and definably compact. (In the o-minimal case this is Fact 3.25 of ). Finally, we also isolate the definably amenable part of G, which we call the definably amenable component. Note that dfg stands for “has a definable f-generic type", and fsg for “has finitely satisfiable generics", which will be discussed together with various equivalences. We will need to understand something about groups of the form G(k) where k is a p-adically closed field and G a semisimple algebraic group over k, and as part of the analysis we will prove the Kneser-Tits conjecture over p-adically closed fields.

We study finite-dimensional groups definable in models of the theory of real closed fields with a generic derivation (also known as CODF). We prove that any such group definably embeds in a semialgebraic group. We extend the results to several more general contexts; strongly model complete theories of large geometric fields with a generic derivation, model complete o-minimal expansions of RCF with a generic derivation, open theories of topological fields with a generic derivation. We also give a general theorem on recovering a definable group from generic data in the context of geometric structures.

The parameter identifiability problem for a dynamical system is to determine whether the parameters of the system can be found from data for the outputs of the system. Verifying whether the parameters are identifiable is a necessary first step before a meaningful parameter estimation can take place. Non-identifiability occurs in practical models. To reparametrize a model to achieve identifiability is a challenge. The existing approaches have been shown to be useful for many important examples. However, these approaches are either limited to linear models and scaling parametrizations or are not guaranteed to find a reparametrization even if it exists. In the present paper, we prove that there always exists a locally identifiable model with the same input-output behaviour as the original one obtained from a given one by a partial specialization of the parameters. Furthermore, we give a sufficient observability condition for the existence of a state space transformation from the original model to the new one. Our proof is constructive and can be translated to an algorithm, which we illustrate by several examples.

We first give simplified and corrected accounts of some results in work by Pillay (2017) on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing (1938) to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing’s work, the Jordan–Schur theorem, and a (relatively) more recent result of Kazhdan (1982) on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudoamenable groups to compact Lie groups. Together with the stabilizer theorems of Hrushovski (2012) and Montenegro et al. (2020), we obtain a uniform (but non-quantitative) analogue of Bogolyubov’s lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.

In \cite{Pillay} and more formally in \cite{Onshuus-Pillay} it was asked whether open subgroups of $p$-adic algebraic groups are ($p$-adic) semialgebraic, equivalently, definable in the structure $(\mathbb Q_{p}, +, \times)$. We give a positive answer in the commutative case. Together with results of \cite{Prasad} this leads to a positive answer for reductive algebraic groups.

We adapt the notion from [7] and [2] of a (relatively) definable subset of Aut(M) when M is a saturated structure, to the case Aut(M/A) when M is atomic and strongly omega-homogeneous (over a set A). We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of Aut(M/A). For example when T is stable, we have existence and uniqueness. We also discuss the compatibility of our definability notions with definable Galois cohomology from [12] and differential Galois theory. (c) 2025 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

We show that the free module of infinite rank R(kappa) purely embeds every t-generated flat left R-module iff R is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory T of R( kappa) whose projectivity is equivalent to left perfectness, which allows to add a 'stronger' equivalent condition: R(kappa) purely embeds every t-generated flat left Rmodule which is a model of T. We extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a 'Bass theory' of pure-projective modules. We put this new theory to use by, among other things, reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Throughout, T denotes a complete first-order theory in a countable language L that has infinite models and I(ℵ_0,T) denotes the number of countable models of T, up to an isomorphism. To determine I(ℵ_0,T), it suffices to consider only countable models of T with domain ω; since there are at most continuum many L-structures with domain ω, I(ℵ_0,T)⩽ 2^ℵ_0 holds. Theories with I(ℵ_0,T)=1 are the ℵ_0-categorical theories. These include the theory of an infinite set, theories of infinite-dimensional vector spaces over a finite field, and the theory of dense linear orders. Theories with I(ℵ_0,T)<2^ℵ_0 are said to have few countable models. In this paper we discuss and survey work done on Vaught's conjecture, Martin's conjecture, and Ehhrenfeuch theories (theories with more than one but only finitely many, countable models).

We investigate the question of when free structures of infinite rank (in a variety) possess model-theoretic properties like categoricity in higher power, saturation, or universality. Concentrating on left R-modules we show, among other things, that the free module of infinite rank R^(κ) embeds every κ-generated flat left R-module iff R is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory T of R^(κ) whose projectivity is equivalent to left perfectness, which allows to add a "stronger" equivalent condition: R^(κ) embeds every κ-generated flat left R-module which is a model of T. In addition, we extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of pure-projective modules. We put this new theory to use by reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules.

Let K be differential field with algebraically closed field of constants. Let K^diff be a differential closure of K, and L the (iterated) Picard-Vessiot closure of K inside K^diff. Let G be a linear differential algebraic group over K and X a differential algebraic torsor for G over K. We prove that X(L) is Kolchin-dense in X. When G is finite-dimensional we prove that X(L) = X(K^diff). We give close relationships between Picard-Vessiot extensions of K and torsors for suitable finite-dimensional linear differential algebraic groups over K. We suggest some differential field analogues of the notion of boundedness for fields (Serre's property F).

We give an 'arithmetic regularity lemma' for groups definable in finite fields, analogous to Tao's 'algebraic regularity lemma' for graphs definable in finite fields. More specifically, we show that, for any M>0, any finite field 𝐅, and any definable group (G,·) in 𝐅 and definable subset D⊆ G, each of complexity at most M, there is a normal definable subgroup H⩽ G, of index and complexity O_M(1), such that the following holds: for any cosets V,W of H, the bipartite graph (V,W,xy^-1∈ D) is O_M(|𝐅|^-1/2)-quasirandom. Various analogous regularity conditions follow; for example, for any g∈ G, the Fourier coefficient ||1_H∩ Dg(π)||_op is O_M(|𝐅|^-1/8) for every non-trivial irreducible representation π of H.