O-minimal Research Articles

Neural network learning and Quantum Gravity

The landscape of low-energy effective field theories stemming from string theory is too vast for a systematic exploration. However, the meadows of the string landscape may be fertile ground for the application of machine learning techniques. Employing neural network learning may allow for inferring novel, undiscovered properties that consistent theories in the landscape should possess, or checking conjectural statements about alleged characteristics thereof. The aim of this work is to describe to what extent the string landscape can be explored with neural network-based learning. Our analysis is motivated by recent studies that show that the string landscape is characterized by finiteness properties, emerging from its underlying tame, o-minimal structures. Indeed, employing these results, we illustrate that any low-energy effective theory of string theory is endowed with certain statistical learnability properties. Consequently, several local learning problems therein formulated, including interpolations and multi-class classification problems, can be concretely addressed with machine learning, delivering results with sufficiently high accuracy.

Complexity in tame quantum theories

Inspired by the notion that physical systems can contain only a finite amount of information or complexity, we introduce a framework that allows for quantifying the amount of logical information needed to specify a function or set. We then apply this methodology to a variety of physical systems and derive the complexity of parameter-dependent physical observables and coupling functions appearing in effective Lagrangians. In order to implement these ideas, it is essential to consider physical theories that can be defined in an o-minimal structure. O-minimality, a concept from mathematical logic, encapsulates a tameness principle. It was recently argued that this property is inherent to many known quantum field theories and is linked to the UV completion of the theory. To assign a complexity to each statement in these theories one has to further constrain the allowed o-minimal structures. To exemplify this, we show that many physical systems can be formulated using Pfaffian o-minimal structures, which have a well-established notion of complexity. More generally, we propose adopting sharply o-minimal structures, recently introduced by Binyamini and Novikov, as an overarching framework to measure complexity in quantum theories.

Non-Essential Expansions of Quite o-Minimal Theories

Non-Essential Expansions of Quite o-Minimal Theories

Nondefinability results with entire functions of finite order in polynomially bounded o-minimal structures

Nondefinability results with entire functions of finite order in polynomially bounded o-minimal structures

Вариации жесткости для упорядоченных теорий

One of the important characteristics of structures is degrees of semantic and syntactic rigidity, as well as indices of rigidity, showing how much the given structure differs from semantically rigid structures, i.e., structures with one-element automorphism groups, as well as syntactically rigid structures, i.e., structures covered by definable closure of the empty set. Issues of describing the degrees and indices of rigidity represents interest both in a general context and in relation to ordering theories and their models. In the given paper, we study possibilities for semantic and syntactic rigidity for ordered theories, i.e., the rigidity with respect to automorphism group and with respect to definable closure. We describe values for indices and degrees of semantic and syntactic rigidity for well-ordered sets, for discrete, dense, and mixed orders and for countable models of $\aleph_0$-categorical weakly o-minimal theories. All possibilities for degrees of rigidity for countable linear orderings are described.

A non-archimedean definable Chow theorem

Peterzil and Starchenko have proved the following surprising generalization of Chow’s theorem: A closed analytic subset of a complex algebraic variety that is definable in an o-minimal structure is in fact an algebraic subset. In this paper, we prove a non-archimedean analogue of this result.

UNDEFINABILITY OF MULTIPLICATION IN PRESBURGER ARITHMETIC WITH SETS OF POWERS

Abstract We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number of sets of powers does not define multiplication.

ORTHOGONAL DECOMPOSITION OF DEFINABLE GROUPS

Abstract Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. As an application, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures.

Definable Version of Wedderburn–Artin Theorem in O-Minimal Structures

Definable Version of Wedderburn–Artin Theorem in O-Minimal Structures

Mini-Workshop: Topological and Differential Expansions of o-minimal Structures

The workshop brought together researchers with expertise in areas of mathematics where model theory has had interesting applications. The areas of expertise spanned from expansions of o-minimal structures preserving tame geometric properties to expansions of specified fields by classical operators that preserve neo-stability properties. There were presentations and discussions on recent developments in definable groups and decompositions in relatively tame setups, the interplay of different notions of dimension and closure operators, and applications of the model theory of differential fields to diophantine geometry.

Companionability characterization for the expansion of an o-minimal theory by a dense subgroup

This paper provides a full characterization for when the expansion of a complete o-minimal theory, one that extends the theory of ordered divisible abelian groups, by a unary predicate that picks out a divisible, dense and codense group has a model companion. This result is motivated by criteria and questions introduced in the recent works [14] and [10] concerning the existence of model companions, as well as preservation results for some neostability properties when passing to the model companion. Examples are included both in which the predicate is an additive subgroup of a real ordered vector space, and where it is a multiplicative subgroup of the nonzero elements of an o-minimal expansion of a real closed field. The paper concludes with a brief discussion of neostability properties and examples that illustrate the lack of preservation (from the base o-minimal theory to the model companion of the expansion we define) for properties such as strong, NIP, and NTP2, though there are also examples for which some or all three of those properties are preserved.

PROPERTIES OF HYPERGRAPHS OF MODELS OF WEAKLY O-MINIMAL THEORIES

In this paper, we study the notions of relative H-freedom and relative H-independence for hypergraphs of models of weakly o-minimal theories. Hypergraphs of models of a theory are derived objects that allow obtaining essential structural information both about the theories themselves and about related semantic objects. Recall that a hypergraph is any pair of sets (X, Y), where Y is some subset of the Boolean P(X) of a set X. In this case, the set X is called the support of the hypergraph (X, Y), and elements from Y are called edges of the hypergraph (X, Y). Weak o-minimality was originally deeply investigated by D. Macpherson, D. Marker, and C. Steinhorn. In the nineties of the last century, Kazakhstan scientists successfully joined the study of this concept, solving a number of problems posed by the authors. In this paper, we continue the study of model-theoretic properties of weakly o-minimal structures. A criterion for relative H-freedom of the set of realizations of non-algebraic 1-type in almost omega-categorical weakly o-minimal theories is obtained in terms of convexity rank. We also establish a criterion for relative H-independence of the sets of realizations of two non-algebraic 1-types in almost omega-categorical weakly o-minimal theories in terms of weak orthogonality of 1-types.

Correction: Global convergence of the gradient method for functions definable in o-minimal structures

Correction: Global convergence of the gradient method for functions definable in o-minimal structures

On preparation theorems for R<sub>an, exp</sub>-definable functions

In this article we give strong versions for preparation theorems for $\mathbb{R}_{\textnormal{an,exp}}$-definable functions outgoing from methods of Lion and Rolin ($\mathbb{R}_{\textnormal{an,exp}}$ is the o-minimal structure generated by all restricted analytic functions and the global exponential function). By a deep model theoretic fact of Van den Dries, Macintyre and Marker every $\mathbb{R}_{\textnormal{an,exp}}$-definable function is piecewise given by $\mathcal{L}_{\textnormal{an}}(\exp,\log)$-terms where $\mathcal{L}_{\textnormal{an}}(\exp,\log)$ denotes the language of ordered rings augmented by all restricted analytic functions, the global exponential and the global logarithm. So our idea is to consider log-analytic functions at first, i.e. functions which are iterated compositions from either side of globally subanalytic functions and the global logarithm, and then $\mathbb{R}_{\textnormal{an,exp}}$-definable functions as compositions of log-analytic functions and the global exponential.

Finiteness for self-dual classes in integral variations of Hodge structure

We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$.

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We show how to derive triangulations of sets locally definable in o-minimal structures from triangulations of compact definable sets. We give it in particular for strict Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal C^p$$\\end{document}-triangulations which has been recently studied by the author. This combined with a theorem of Fernando and Ghiloni implies that every continuous mapping defined on a locally compact subset B of Rm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R^m$$\\end{document} with values in any locally definable and locally compact subset A of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R^n$$\\end{document} can be approximated by Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal C^p$$\\end{document}-mappings defined on B with values in A for any positive integer p.

Strict $\mathcal{C}^p$-triangulations – a new approach to desingularization

Let R be any real closed field expanded by some o-minimal structure. Let f: A\to R^d be a definable and continuous mapping defined on a definable, closed, bounded subset A of R^n . Let \mathcal{E} be a finite family of definable subsets of R^n contained in A . Let p be any positive integer. We prove that then there exists a finite simplicial complex \mathcal{T} in R^n and a definable homeomorphism h: |\mathcal{T} |\to A , where |\mathcal{T}|:= \bigcup \mathcal{T} , such that for each simplex \varDelta\in \mathcal{T} , the restriction of h to its relative interior \mathring{\varDelta} is a \mathcal{C}^p -embedding of \mathring{\varDelta} into R^n and moreover both h and f\circ h are of class \mathcal{C}^p in the sense that they have definable \mathcal{C}^p -extensions defined on an open definable neighborhood of | \mathcal{T}| in R^n . We then call a pair ( \mathcal{T}, h) a strict \mathcal{C}^p -triangulation of A . In addition, this triangulation can be made compatible with \mathcal{E} in the sense that for each E\in \mathcal{E} , h^{-1}(E) is a union of some \mathring{\varDelta} , where \varDelta\in \mathcal{T} . We also give an application to approximation theory.

On minimal flows and definable amenability in some distal NIP theories

We study the definable topological dynamics (G(M),SG(M)) of a definable group acting on its type space, where M is either an o-minimal structure or a p-adically closed field, and G a definably amenable group. We focus on the problem raised in [13] of whether weakly generic types coincide with almost periodic types, showing that the answer is positive when G has boundedly many global weakly generic types. We also give two “minimal counterexamples” where G has unboundedly many global weakly generic types, extending the main results in [21] to a more general context.

ALMOST 1-TRANSITIVITY IN LINEARLY ORDERED STRUCTURES

The present paper concerns the notion of weak o-minimality introduced by M. Dickmann and originally deeply studied by D. Macpherson, D. Marker, and C. Steinhorn. Weak o-minimality is a generalization of the notion of o-minimality introduced by A. Pillay and C. Steinhorn in series of joint papers. As is known, the ordered field of real numbers is an example of an o-minimal structure. We continue studying model-theoretic properties of o-minimal and weakly o-minimal structures. In particular, we introduce the notion of almost 1-transitivity in linearly ordered structures and study tits properties. Almost 1-transitive o-minimal and weakly o-minimal linear orderings have been described. It has been established that an almost 1-transitive weakly o-minimal linear ordering is isomorphic to a finite number of concatenations of almost 1-transitive o-minimal linear orderings. Properties of expansions of families of almost 1-transitive linearly ordered theories are studied. Rank values for families of almost 1-transitive o-minimal and weakly o-minimal linear orderings have been found. A criterion for preserving both the almost 1-transitivity and weak o-minimality has been found at expanding an almost 1-transitive weak o-minimal theory by an arbitrary unary predicate. Dense ordering of an almost 1-transitive weakly o-minimal theory that is almost omega-categorical has been established.

Liouville closed HT-fields

Let T be a complete, model complete o-minimal theory extending the theory of real closed ordered fields. An HT-field is a model K of T equipped with a T-derivation ▪ such that the underlying ordered differential field of ▪ is an H-field. We study HT-fields and their extensions. Our main result is that if T is power bounded, then every HT-field K has either exactly one or exactly two minimal Liouville closed HT-field extensions up to K-isomorphism. The assumption of power boundedness can be relaxed to allow for certain exponential cases, such as T=Th(Ran,exp).