arXiv subjects

Paul Z. Wang

Publications and source records attributed to Paul Z. Wang.

5 recordsLinked to original sources

arXiv2026

Separation and Gluing of Explanations on Sites of Dynamical Systems

We construct a Grothendieck site whose objects are Mealy machines over definable sets in an o-minimal structure and whose coverings are jointly surjective families of definable open immersions. On this site, we define presheaves of explanations -- systems equipped with an interpretable interface, parameterised by a ``judge.'' We prove that the behavioral presheaf (quotienting by observable output equivalence) is separated: a global explanation is determined by its local restrictions. We show that gluing fails in general -- locally consistent explanations need not assemble globally -- and give, for stateless explanatory systems of the restricted-interface presheaf, a necessary and sufficient topological condition for the sheaf property in terms of robust disconnection of fibers of the judge.

math.CT↗

arXiv2025

Residually Dominated Groups in Henselian Valued Fields of Equicharacteristic Zero

We introduce \emph{residually dominated groups} in pure henselian valued fields of equicharacteristic zero, as an analogue of stably dominated groups introduced by Hrushovski and Rideau-Kikuchi. We show that when $G$ is a residually dominated group, there is a finite-to-one group homomorphism from its connected component into a connected stably dominated group, and we study the functoriality and universality properties of this map. Moreover, we prove that residual domination is witnessed by a group homomorphism into a definable group in the residue field. In our proofs, we use the results of Montenegro, Onshuus, and Simon on groups definable in $\mathrm{NTP}_2$-theories that extend the theory of fields. Along the way, we also provide an algebraic characterization of residually dominated types, generalizing the work by Ealy, Haskell and Simon for stably dominated types in algebraically closed valued fields, and we study their properties.

math.LO↗

arXiv2025

On groups and fields interpretable in $\mathrm{NTP_2}$ fields

This paper aims at developing model-theoretic tools to study interpretable fields and definably amenable groups, mainly in $\mathrm{NIP}$ or $\mathrm{NTP_2}$ settings. An abstract theorem constructing definable group homomorphisms from generic data is proved. It relies heavily on a stabilizer theorem of Montenegro, Onshuus and Simon. The main application is a structure theorem for definably amenable groups that are interpretable in algebraically bounded perfect $\mathrm{NTP_2}$ fields with bounded Galois group (under some mild assumption on the imaginaries involved), or in algebraically bounded theories of (differential) NIP fields. These imply a classification of the fields interpretable in differentially closed valued fields, and structure theorems for fields interpretable in finitely ramified henselian valued fields of characteristic $0$, or in NIP algebraically bounded differential fields.

math.LO↗

arXiv2024

Extending Hrushovski's groupoid-cover correspondence using simplicial groupoids

Hrushovski's suggestion, given in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics , 2012], to capture the structure of the 1-analysable covers of a theory T using simplicial groupoids definable in T is realized here. The ideas of Haykazyan and Moosa, found in ["Functoriality and uniformity in Hrushovski's groupoid-cover correspondence," Annals of Pure and Applied Logic , 2018] are used, and extended, to define an equivalence of categories. Finally, a couple of examples are studied with these new tools.

math.LO↗

arXiv2023

The group configuration theorem for generically stable types

We generalize Hrushovski's group configuration theorem to the case where the type of the configuration is generically stable, without assuming tameness of the ambient theory. The properties of generically stable types, which we recall in the first section, enable us to adapt the proof known in the stable context.

math.LO↗