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Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics (QM) that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, ‘Q-worlds’. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.
We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A3| ≤ O(|A|), or small alternation, |AA−1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.
We present three examples of countable homogeneous structures (also called Fraïssé limits) whose automorphism groups are not universal, namely, fail to contain isomorphic copies of all automorphism groups of their substructures.
Our first example is a particular case of a rather general construction on Fraïssé classes, which we call diversification, leading to automorphism groups containing copies of all finite groups. Our second example is a special case of another general construction on Fraïssé classes, the mixed sums, leading to a Fraïssé class with all finite symmetric groups appearing as automorphism groups and at the same time with a torsion-free automorphism group of its Fraïssé limit. Our last example is a Fraïssé class of finite models with arbitrarily large finite abelian automorphism groups, such that the automorphism group of its Fraïssé limit is again torsion-free.
It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with ‘solubility’ replaced by ‘nilpotence’ and ‘perfectness’, among others, are false.
These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini’s theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.
A first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of the theory of proper extensions of algebraically closed fields and of the theory of separably closed fields of arbitrary imperfection degree.
There is a Turing computable embedding
$\Phi $
of directed graphs
$\mathcal {A}$
in undirected graphs (see [15]). Moreover, there is a fixed tuple of formulas that give a uniform effective interpretation; i.e., for all directed graphs
$\mathcal {A}$
, these formulas interpret
$\mathcal {A}$
in
$\Phi (\mathcal {A})$
. It follows that
$\mathcal {A}$
is Medvedev reducible to
$\Phi (\mathcal {A})$
uniformly; i.e.,
$\mathcal {A}\leq _s\Phi (\mathcal {A})$
with a fixed Turing operator that serves for all
$\mathcal {A}$
. We observe that there is a graph G that is not Medvedev reducible to any linear ordering. Hence, G is not effectively interpreted in any linear ordering. Similarly, there is a graph that is not interpreted in any linear ordering using computable
$\Sigma _2$
formulas. Any graph can be interpreted in a linear ordering using computable
$\Sigma _3$
formulas. Friedman and Stanley [4] gave a Turing computable embedding L of directed graphs in linear orderings. We show that there is no fixed tuple of
$L_{\omega _1\omega }$
-formulas that, for all G, interpret the input graph G in the output linear ordering
$L(G)$
. Harrison-Trainor and Montalbán [7] have also shown this, by a quite different proof.
We characterize the linear order types
$\tau $
with the property that given any countable linear order
$\mathcal {L}$
,
$\tau \cdot \mathcal {L}$
is a computable linear order iff
$\mathcal {L}$
is a computable linear order, as exactly the finite nonempty order types.
Many phenomena in geometry and analysis can be explained via the theory of
$D$
-modules, but this theory explains close to nothing in the non-archimedean case, by the absence of integration by parts. Hence there is a need to look for alternatives. A central example of a notion based on the theory of
$D$
-modules is the notion of holonomic distributions. We study two recent alternatives of this notion in the context of distributions on non-archimedean local fields, namely
$\mathscr{C}^{\text{exp}}$
-class distributions from Cluckers et al. [‘Distributions and wave front sets in the uniform nonarchimedean setting’, Trans. Lond. Math. Soc.5(1) (2018), 97–131] and WF-holonomicity from Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)]. We answer a question from Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)] by showing that each distribution of the
$\mathscr{C}^{\text{exp}}$
-class is WF-holonomic and thus provides a framework of WF-holonomic distributions, which is stable under taking Fourier transforms. This is interesting because the
$\mathscr{C}^{\text{exp}}$
-class contains many natural distributions, in particular, the distributions studied by Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)]. We show also another stability result of this class, namely, one can regularize distributions without leaving the
$\mathscr{C}^{\text{exp}}$
-class. We strengthen a link from Cluckers et al. [‘Distributions and wave front sets in the uniform nonarchimedean setting’, Trans. Lond. Math. Soc.5(1) (2018), 97–131] between zero loci and smooth loci for functions and distributions of the
$\mathscr{C}^{\text{exp}}$
-class. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory.
We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are C-minimal.
We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: we construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.
In this article, we functorially associate definable sets to
$k$
-analytic curves, and definable maps to analytic morphisms between them, for a large class of
$k$
-analytic curves. Given a
$k$
-analytic curve
$X$
, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of
$X$
and show that they satisfy a bijective relation with the radial subsets of
$X$
. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of
$k$
-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser’s theorem on iso-definability of curves. However, our approach can also be applied to strictly
$k$
-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting.
We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated with sequences such as
$(-n^{s})_{n>0}$
(for
$s>0$
) and
$(-s^{n})_{n>0}$
(for
$s>1$
), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known so far: (i) the expansion is o-minimal (that is, definable sets have only finitely many connected components); (ii) every Borel subset of each
$\mathbb{R}^{n}$
is definable; (iii) the expansion is interdefinable with a structure of the form
$(\mathfrak{R}^{\prime },\unicode[STIX]{x1D6FC}^{\mathbb{Z}})$
where
$\unicode[STIX]{x1D6FC}>1$
,
$\unicode[STIX]{x1D6FC}^{\mathbb{Z}}$
is the set of all integer powers of
$\unicode[STIX]{x1D6FC}$
, and
$\mathfrak{R}^{\prime }$
is o-minimal and defines no irrational power functions.
Let
$T_{n}(\mathbb{F})$
be the semigroup of all upper triangular
$n\times n$
matrices over a field
$\mathbb{F}$
. Let
$UT_{n}(\mathbb{F})$
and
$UT_{n}^{\pm 1}(\mathbb{F})$
be subsemigroups of
$T_{n}(\mathbb{F})$
, respectively, having
$0$
s and/or
$1$
s on the main diagonal and
$0$
s and/or
$\pm 1$
s on the main diagonal. We give some sufficient conditions under which an involution semigroup is nonfinitely based. As an application, we show that
$UT_{2}(\mathbb{F}),UT_{2}^{\pm 1}(\mathbb{F})$
and
$T_{2}(\mathbb{F})$
as involution semigroups under the skew transposition are nonfinitely based for any field
$\mathbb{F}$
.
We establish the first global results for groups definable in tame expansions of o-minimal structures. Let
${\mathcal{N}}$
be an expansion of an o-minimal structure
${\mathcal{M}}$
that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of
${\mathcal{M}}$
by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil’s group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if
${\mathcal{N}}$
expands
${\mathcal{M}}$
by a dense independent set, then every definable group is o-minimal.
We adapt the classical notion of building models by games to the setting of continuous model theory. As an application, we study to what extent canonical operator algebras are enforceable models. For example, we show that the hyperfinite II1 factor is an enforceable II1 factor if and only if the Connes Embedding Problem has a positive solution. We also show that the set of continuous functions on the pseudoarc is an enforceable model of the theory of unital, projectionless, abelian $\text{C}^{\ast }$-algebras and use this to show that it is the prime model of its theory.
We investigate bounds in Ramsey’s theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek (Duke Mathematical Journal163(12) (2014), 2243–2270) from the semialgebraic case to arbitrary polynomially bounded $o$-minimal expansions of $\mathbb{R}$, and show that it does not hold in $\mathbb{R}_{\exp }$. This provides a new combinatorial characterization of polynomial boundedness for $o$-minimal structures. We also prove an analog for relations definable in $P$-minimal structures, in particular for the field of the $p$-adics. Generalizing Conlon et al. (Transactions of the American Mathematical Society366(9) (2014), 5043–5065), we show that in distal structures the upper bound for $k$-ary definable relations is given by the exponential tower of height $k-1$.
We give a geometric and elementary proof of the uniform
$\mathscr{C}^{p}$
-parametrization theorem of Yomdin and Gromov in arbitrary o-minimal structures.
In unpublished notes, Pila discussed some theory surrounding the modular function j and its derivatives. A focal point of these notes was the statement of two conjectures regarding j, j′ and j″: a Zilber–Pink-type statement incorporating j, j′ and j″, which was an extension of an apparently weaker conjecture of André–Oort type. In this paper, I first cover some background regarding j, j′ and j″, mostly covering the work already done by Pila. Then I use a seemingly novel adaptation of the o-minimal Pila–Zannier strategy to prove a weakened version of Pila's ‘Modular André–Oort with Derivatives’ conjecture. Under the assumption of a certain number-theoretic conjecture, the central theorem of the paper implies Pila's conjecture in full generality, as well as a more precise statement along the same lines.
We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function
$f(x)$
in a quasianalytic Denjoy–Carleman class
${\mathcal{Q}}_{M}$
, which is formally composite with a generically submersive mapping
$y=\unicode[STIX]{x1D711}(x)$
of class
${\mathcal{Q}}_{M}$
, at a single given point in the source (or in the target) of
$\unicode[STIX]{x1D711}$
can be written locally as
$f=g\circ \unicode[STIX]{x1D711}$
, where
$g(y)$
belongs to a shifted Denjoy–Carleman class
${\mathcal{Q}}_{M^{(p)}}$
; (2) a statement on a similar loss of regularity for functions definable in the
$o$
-minimal structure given by expansion of the real field by restricted functions of quasianalytic class
${\mathcal{Q}}_{M}$
. Both results depend on an estimate for the regularity of a
${\mathcal{C}}^{\infty }$
solution
$g$
of the equation
$f=g\circ \unicode[STIX]{x1D711}$
, with
$f$
and
$\unicode[STIX]{x1D711}$
as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.