Related papers: Valued fields, Metastable groups

We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued fields is stably embedded in an elementary extension if and only if its…

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…

Let $K$ be a henselian valued field with ${\cal O}_K$ its valuation ring, $\Gamma$ its value group, and $\boldsymbol{k}$ its residue field. We study the definable subsets of ${\cal O}_K$ and algebraic groups definable over ${\cal O}_K$ in…

A binding group theorem is proved in the context of quantifier-free internality to the fixed field in difference-closed fields of characteristic zero. This is articulated as a statement about the birational geometry of isotrivial algebraic…

We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field,…

For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group,…

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a…

We give a complete list of the one-dimensional groups definable in algebraically closed valued fields and i the pseudo-local fields, up to a finite index subgroup and a quotient by a finite subgroup.

We continue the effort of grokking the structure of power-bounded $T$-convex valued fields, whose theory is in general referred to as TCVF. In the present paper our focus is on certain expansion of it that is equipped with a tempered…

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble…

The aim of this paper is to develop the theory of groups definable in the $p$-adic field ${\mathbb Q}_p$, with ``definable $f$-generics" in the sense of an ambient saturated elementary extension of ${\mathbb Q}_p$. We call such groups…

Let p be a fixed prime. An Abelian p-group is an Abelian group (not necessarily finitely generated) in which every element has for its order some power of p. The countable Abelian p-groups are classified by Ulm's theorem, and Khisamiev…

We show that separably closed valued fields of finite imperfection degree (either with lambda-functions or commuting Hasse derivations) eliminate imaginaries in the geometric language. We then use this classification of interpretable sets…

We study the model theory of the $2$-sorted structure $(\mathbb{F}, \mathbb{C};\chi)$, where $\mathbb{F}$ is an algebraic closure of a finite field of characteristic $p$, $\mathbb{C}$ is the field of complex numbers and $\chi: \mathbb{F}…

A definable set $X$ in the first-order language of rings defines a family of random vectors: for each finite field $\mathbb{F}_q$, let the distribution be supported and uniform on the $\mathbb{F}_q$-rational points of $X$. We employ results…

We prove that the class of separably algebraically closed valued fields equipped with a distinguished Frobenius endomorphism $x \mapsto x^q$ is decidable, uniformly in $q$. The result is a simultaneous generalization of the work of…

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…

We survey various approaches to axiomatic stable homotopy theory, with examples including derived categories, categories of (possibly equivariant or localized) spectra, and stable categories of modular representations of finite groups. We…

Let $V$ be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue $\hat {V}$ of the Berkovich analytification $V^{an}$ of $V$,…

We present a new proof of descent for stably dominated types in any theory, dropping the hypothesis of the existence of global invariant extensions. Additionally, we give a much simpler proof of descent for stably dominated types in…