arXiv subjects

James Hanson

Publications and source records attributed to James Hanson.

15 recordsLinked to original sources

arXiv2023

A simple continuous theory

In the context of continuous first-order logic, special attention is often given to theories that are somehow continuous in an 'essential' way. A common feature of such theories is that they do not interpret any infinite discrete structures. We investigate a stronger condition that is easier to establish and use it to give an example of a strictly simple continuous theory that does not interpret any infinite discrete structures: the theory of richly branching $\mathbb{R}$-forests with generic binary predicates. We also give an example of a superstable theory that fails to satisfy this stronger condition but nevertheless does not interpret any infinite discrete structures.

math.LO↗

arXiv2023

A metric set theory with a universal set

Motivated by ideas from the model theory of metric structures, we introduce a metric set theory, $\mathsf{MSE}$, which takes bounded quantification as primitive and consists of a natural metric extensionality axiom (the distance between two sets is the Hausdorff distance between their extensions) and an approximate, non-deterministic form of full comprehension (for any real-valued formula $\varphi(x,y)$, tuple of parameters $a$, and $r < s$, there is a set containing the class $\{x: \varphi(x,a) \leq r\}$ and contained in the class $\{x:\varphi(x,a) < s\}$). We show that $\mathsf{MSE}$ is sufficient to develop classical mathematics after the addition of an appropriate axiom of infinity. We then construct canonical representatives of well-order types and prove that ultrametric models of $\mathsf{MSE}$ always contain externally ill-founded ordinals, conjecturing that this is true of all models. To establish several independence results and, in particular, consistency, we construct a variety of models, including pseudo-finite models and models containing arbitrarily large standard ordinals. Finally, we discuss how to formalize $\mathsf{MSE}$ in either continuous logic or {\L}ukasiewicz logic.

math.LO↗

arXiv2023

Some semilattices of definable sets in continuous logic

In continuous first-order logic, the union of definable sets is definable but generally the intersection is not. This means that in any continuous theory, the collection of $\varnothing$-definable sets in one variable forms a join-semilattice under inclusion that may fail to be a lattice. We investigate the question of which semilattices arise as the collection of definable sets in a continuous theory. We show that for any non-trivial finite semilattice $L$ (or, equivalently, any finite lattice $L$), there is a superstable theory $T$ whose semilattice of definable sets is $L$. We then extend this construction to some infinite semilattices. In particular, we show that the following semilattices arise in continuous theories: $\alpha+1$ and $(\alpha+1)^\ast$ for any ordinal $\alpha$, a semilattice containing an exact pair above $\omega$, and the lattice of filters in $L$ for any countable meet-semilattice $L$. By previous work of the author, this establishes that these semilattices arise in stable theories. The first two are done in languages of cardinality $\aleph_0 + |\alpha|$, and the latter two are done in countable languages.

math.LO↗

arXiv2022

Approximate isomorphism of randomization pairs

We study approximate $\aleph_0$-categoricity of theories of beautiful pairs of randomizations, in the sense of continuous logic. This leads us to disprove a conjecture of Ben Yaacov, Berenstein and Henson, by exhibiting $\aleph_0$-categorical, $\aleph_0$-stable metric theories $Q$ for which the corresponding theory $Q_P$ of beautiful pairs is not approximately $\aleph_0$-categorical, i.e., has separable models that are not isomorphic even up to small perturbations of the smaller model of the pair. The theory $Q$ of randomized infinite vector spaces over a finite field is such an example. On the positive side, we show that the theory of beautiful pairs of randomized infinite sets is approximately $\aleph_0$-categorical. We also prove that a related stronger property, which holds in that case, is stable under various natural constructions, and formulate our guesswork for the general case.

math.LO↗

arXiv2022

Bounded ultraimaginary independence and its total Morley sequences

We investigate the following model-theoretic independence relation: $\def\indbu{{\rlap{\hspace11.9mu\vert}\lower7.5mu\smile}^{\!\mathrm{bu}}} b \indbu_A\hspace3mu c$ if and only if $\mathrm{bdd}^u(Ab)\cap \mathrm{bdd}^u(Ac) = \mathrm{bdd}^u(A)$, where $\mathrm{bdd}^u(X)$ is the class of all ultraimaginaries bounded over $X$. In particular, we sharpen a result of Wagner to show that $b \indbu_A\hspace3mu c$ if and only if $\langle \mathrm{Autf}(\mathbb{M}/Ab)\cup\mathrm{Autf}(\mathbb{M}/Ac) \rangle = \mathrm{Autf}(\mathbb{M}/A)$, and we establish full existence over hyperimaginary parameters (i.e., for any set of hyperimaginaries $A$ and ultraimaginaries $b$ and $c$, there is a $b' \equiv_A b$ such that $b' \indbu_A\hspace3mu c$). Extension then follows as an immediate corollary. We also study total $\hspace-5mu\indbu$-Morley sequences (i.e., $A$-indiscernible sequences $I$ satisfying $J \indbu_A\hspace3mu K$ for any $J$ and $K$ with $J + K \equiv^{\mathrm{EM}}_A I$), and we prove that an $A$-indiscernible sequence $I$ is a total $\hspace-5mu\indbu$-Morley sequence over $A$ if and only if whenever $I$ and $I'$ have the same Lascar strong type over $A$, $I$ and $I'$ are related by the transitive, symmetric closure of the relation '$J+K$ is $A$-indiscernible.' This is also equivalent to $I$ being 'based on' $A$ in a sense defined by Shelah in his early study of simple unstable theories. Finally, we show that for any $A$ and $b$ in any theory $T$, if there is an Erd\"os cardinal $\kappa(\alpha)$ with $|Ab|+|T| < \kappa(\alpha)$, then there is a total $\hspace-5mu\indbu$-Morley sequence $(b_i)_{i<\omega}$ over $A$ with $b_0 = b$.

math.LO↗

arXiv2021

Separation for isometric group actions and hyperimaginary independence

We generalize P. M. Neumann's Lemma to the setting of isometric actions on metric spaces and use it to prove several results in continuous logic related to algebraic independence. In particular, we show that algebraic independence satisfies the full existence axiom (which answers a question of Goldbring) and is implied by dividing independence. We also use the relationship between hyperimaginaries and continuous imaginaries to derive further results that are new even for discrete theories. Specifically, we show that if $\mathbb{M}$ is a monster model of a discrete or continuous theory, then bounded-closure independence in $\mathbb{M}^{\text{heq}}$ satisfies full existence (which answers a question of Adler) and is implied by dividing independence.

math.LO↗

arXiv2021

Topometric characterization of type spaces in continuous logic

We show that a topometric space $X$ is topometrically isomorphic to a type space of some continuous first-order theory if and only if $X$ is compact and has an open metric (i.e., satisfies that $\{p : d(p,U) < \varepsilon\}$ is open for every open $U$ and $\varepsilon > 0$). Furthermore, we show that this can always be accomplished with a stable theory.

math.LO↗

arXiv2021

Metric Spaces Are Universal for Bi-interpretation with Metric Structures

In the context of metric structures introduced by Ben Yaacov, Berenstein, Henson, and Usvyatsov, we exhibit an explicit encoding of metric structures in countable signatures as pure metric spaces in the empty signature, showing that such structures are universal for bi-interpretation among metric structures with positive diameter. This is analogous to the classical encoding of arbitrary discrete structures in finite signatures as graphs, but is stronger in certain ways and weaker in others. There are also certain fine grained topological concerns with no analog in the discrete setting.

math.LO↗

arXiv2021

Keisler measures in the wild

We investigate Keisler measures in arbitrary theories. Our initial focus is on Borel definability. We show that when working over countable parameter sets in countable theories, Borel definable measures are closed under Morley products and satisfy associativity. However, we also demonstrate failures of both properties over uncountable parameter sets. In particular, we show that the Morley product of Borel definable types need not be Borel definable (correcting an erroneous result from the literature). We then study various notions of generic stability for Keisler measures and generalize several results from the NIP setting to arbitrary theories. We also prove some positive results for the class of frequency interpretation measures in arbitrary theories, namely, that such measures are closed under convex combinations and commute with all Borel definable measures. Finally, we construct the first example of a complete type which is definable and finitely satisfiable in a small model, but not finitely approximated over any small model.

math.LO↗

arXiv2020

Approximate Isomorphism of Metric Structures

We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov and by Ben Yaacov, Doucha, Nies, and Tsankov, which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two-sorted structures that witness approximate isomorphism. As an application, we show that the theory of any $\mathbb{R}$-tree or ultrametric space of finite radius is stable, improving a result of Carlisle and Henson.

math.LO↗

arXiv2020

Approximate Categoricity in Continuous Logic

We explore approximate categoricity in the context of distortion systems, introduced in our previous paper, which are a mild generalization of perturbation systems, introduced by Ben Yaacov. We extend Ben Yaacov's Ryll-Nardzewski style characterization of separably approximately categorical theories from the context of perturbation systems to that of distortion systems. We also make progress towards an analog of Morley's theorem for inseparable approximate categoricity, showing that if there is some uncountable cardinal $\kappa$ such that every model of size $\kappa$ is 'approximately saturated,' in the appropriate sense, then the same is true for all uncountable cardinalities. Finally we present some examples of these phenomena and highlight an apparent interaction between ordinary separable categoricity and inseparable approximate categoricity.

math.LO↗

arXiv2020

Strongly Minimal Sets and Categoricity in Continuous Logic

The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. Here we investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out. We also introduce, in the context of Banach theories, the notion of an 'indiscernible subspace,' which we use to improve a result of Shelah and Usvyatsov. Both of these notions are applicable to continuous logic outside of the context of inseparably categorical theories. Finally, we construct or present a slew of counterexamples, including an $\omega$-stable theory with no Vaughtian pairs which fails to be inseparably categorical and an inseparably categorical theory with strongly minimal sets in its home sort only over models of sufficiently high dimension.

math.LO↗

arXiv2020

Indiscernible Subspaces and Minimal Wide Types

We develop the machinery of indiscernible subspaces in continuous theories of expansions of Banach spaces, showing that any such theory has an indiscernible subspace and therefore an indiscernible set. We extend a result of Shelah and Usvyatsov by showing that a sequence of realizations of a (possibly unstable) minimal wide type $p$ is a Morley sequence in $p$ if and only if it is the orthonormal basis of an indiscernible subspace in $p$. We also give an example showing that minimal wide types do not generally have type-definable indiscernible subspaces (answering a question of Shelah and Usvyatsov), as well as an example showing that our result fails for non-minimal wide types, even in $\omega$-stable theories.

math.LO↗

arXiv2017

Intersecting D3-D3' system at finite temperature

We analyze the dynamics of intersecting D3/D3' brane system overlapping in 1+1 dimensions, in a holographic treatment where $N$ D3-branes are manifested as anti-de-Sitter Schwartzschild geometry, and the D3'-brane is treated as a probe. We extract the thermodynamic equation of state from the set of embedding solutions, and analyze the stability at the perturbative and the non-perturbative level. We review a systematic procedure to resolve local instabilities and multi-valuedness in the equations of state based on classic ideas of convexity in microcanonical ensumble. We then identify a run-away behavior which was not noticed previously for this system.

hep-th↗

arXiv2015

Dynamics of ${\cal N}=4$ supersymmetric field theories in 2+1 dimensions and their gravity dual

In this note we consider ${\cal N}=4$ SYM theories in 2+1 dimensions with gauge group $U(N)\times U(M)$ and $k$ hypermultiplets charged under the $U(N)$. When $k > 2(N-M)$, the theory flows to a superconformal fixed point in the IR. Theories with $k <2(N-M)$, on the other hand, flows to strong coupling. We explore these theories from the perspective of gravity dual. We find that the gravity duals of theories with $k < (N-M)$ contain enhancons even in situations where repulson singularities are absent. We argue that supergravity description is unreliable in the region near these enhancon points. Instead, we show how to construct reliable sugra duals to particular points on the Coulomb branch where the enhancon is screened. We explore how these singularities reappear as one moves around in Coulomb branch and comment on possible field theory interpretation of this phenomenon. In analyzing gauge/gravity duality for these models, we encountered one unexpected surprise, that the condition for the supergravity solution to be reliable and supersymmetric is somewhat weaker than the expectation from field theory. We also discuss similar issues for theories with $k=0$.

hep-th↗