Rahim Moosa
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Definable Galois theory for bimeromorphic geometry Open
The outlines of a "Galois theory" for bimeromorphic geometry is here developed, via the study of model-theoretic definable binding groups in the theory CCM of compact complex spaces. As an application, a structure theorem about principal m…
Abelian reduction in differential-algebraic and bimeromorphic geometry Open
A new tool for the model theory of differentially closed fields and of compact complex manifolds is here developed. In such settings, it is shown that a type internal to the field of constants (resp. to the projective line) admits a maxima…
View article: FINITE-DIMENSIONAL DIFFERENTIAL-ALGEBRAIC PERMUTATION GROUPS
FINITE-DIMENSIONAL DIFFERENTIAL-ALGEBRAIC PERMUTATION GROUPS Open
Several structural results about permutation groups of finite rank definable in differentially closed fields of characteristic zero (and other similar theories) are obtained. In particular, it is shown that every finite rank definably prim…
A note on subvarieties of powers of OT-manifolds Open
It is shown that the space of finite-to-finite holomorphic correspondences on an OT-manifold is discrete. When the OT-manifold has no proper infinite complex-analytic subsets, it then follows by known model-theoretic results that its carte…
Binding groups for algebraic dynamics Open
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…
View article: Bounding nonminimality and a conjecture of Borovik–Cherlin
Bounding nonminimality and a conjecture of Borovik–Cherlin Open
Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The degree of non…
View article: Finite-dimensional differential-algebraic permutation groups
Finite-dimensional differential-algebraic permutation groups Open
Several structural results about permutation groups of finite rank definable in differentially closed fields of characteristic zero (and other similar theories) are obtained. In particular, it is shown that every finite rank definably prim…
Invariant rational functions under rational transformations Open
Let $X$ be an algebraic variety equipped with a dominant rational self-map $ϕ:X\to X$. A new quantity measuring the interaction of $(X,ϕ)$ with trivial dynamical systems is introduced; the stabilised algebraic dimension of $(X,ϕ)$ captures…
A model theory for meromorphic vector fields Open
Motivated by the study of meromorphic vector fields, a model theory of "compact complex manifolds equipped with a generic derivation" is here proposed. This is made precise by the notion of a differential CCM-structure. A first-order axiom…
View article: BCM volume 65 issue 4 Cover and Front matter
BCM volume 65 issue 4 Cover and Front matter Open
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
A differential analogue of the wild automorphism conjecture Open
A differential analogue of the conjecture of Reichstein, Rogalski, and Zhang in algebraic dynamics is here established: if $X$ is a projective variety over an algebraically closed field of characteristic zero which admits a global algebrai…
Six lectures on model theory and differential-algebraic geometry Open
This is a write-up of some lectures I gave in the Fall of 2021 at the Fields Institute in Toronto, as part of the Thematic Programme on Trends in Pure and Applied Model Theory. The goal of the module was to give a quick introduction to the…
Abelian reduction in differential-algebraic and bimeromorphic geometry Open
A new tool for the model theory of differentially closed fields and of compact complex manifolds is here developed. In such settings, it is shown that a type internal to the field of constants (resp. to the projective line) admits a maxima…
View article: The degree of nonminimality is at most two
The degree of nonminimality is at most two Open
It is shown that if $p$ is a complete type of Lascar rank at least 2 over $A$, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations, $a_1$ and $a_2$, such that $p$ has a nonalgebrai…
Commutative bidifferential algebra Open
Motivated by the Poisson Dixmier-Moeglin equivalence problem, a systematic study of commutative unitary rings equipped with a {\em biderivation}, namely a binary operation that is a derivation in each argument, is here begun, with an eye t…
View article: When any three solutions are independent
When any three solutions are independent Open
Given an algebraic differential equation of order greater than one, it is shown that if there is any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions, along with their derivatives, then there is already s…
View article: BSL volume 27 issue 3 Cover and Front matter
BSL volume 27 issue 3 Cover and Front matter Open
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
View article: Bounding nonminimality and a conjecture of Borovik-Cherlin
Bounding nonminimality and a conjecture of Borovik-Cherlin Open
Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The {\em degree o…
Effective isotrivial Mordell-Lang in positive characteristic Open
The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set $X\capΓ$ when $X$ is a subvariety of a semiabelian variety $G$ over a finite field $\mathbb{F}_q$ and $Γ$ is a finitely generated subgroup of $G$ that is invariant …
INVARIANT HYPERSURFACES Open
The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$ -varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of character…
View article: BSL volume 26 issue 2 Cover and Front matter
BSL volume 26 issue 2 Cover and Front matter Open
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
Model theory and the DME: a survey Open
Recent work using the model theory of differentially closed fields to answer questions having to do with the Dixmier-Moeglin equivalence for (noncommutatve) finitely generated noetherian algebras, and for (commutative) finitely generated P…
-sets and finite automata Open
It is observed that Derksen’s Skolem–Mahler–Lech theorem is a special case of the isotrivial positive characteristic Mordell-Lang theorem due to the second author and Scanlon. This motivates an extension of the classical notion of a -autom…
Internality of logarithmic-differential pullbacks Open
A criterion in the spirit of Rosenlicht is given, on the rational function f(x), for when the planar vector field defined by x'=f(x) and y'=xy admits a pair of algebraically independent first integrals over some extension of the base field…
F-sets and finite automata Open
The classical notion of a k-automatic subset of the natural numbers is here extended to that of an F-automatic subset of an arbitrary finitely generated abelian group $Γ$ equipped with an arbitrary endomorphism F. This is applied to the is…
Isolated types of finite rank: an abstract Dixmier-Moeglin equivalence Open
Suppose $T$ is totally transcendental and every minimal non-locally-modular type is nonorthogonal to a nonisolated minimal type over the empty set. It is shown that a finite rank type $p=tp(a/A)$ is isolated if and only if $a$ is independe…