An invitation to model-theoretic Galois theory

Accessible categories, set theory, and model theory: an invitation

We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality localized to a given category, as well as stable independence, a generalization of pushouts and model-theoretic forking that may interest mathematicians at large. We give many examples, including recently discovered connections with homotopy theory and homological algebra. We also discuss concrete versions of accessible categories (such as abstract elementary cla…

Iterative differential Galois theory: a model theoretic approach

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a G-primitive element theorem holds. In addition, making use of the basic theory of arc spa…

Model theoretic dynamics in Galois fashion

We investigate existentially closed models (of a quite arbitrary theory) equipped which an action of a fixed group G. We embed these structures in a monster model D of some well-rounded theory and describe them as PAC substructures of D. Assuming that the class of these existentially closed models is elementary, we show that, under the assumption of having bounded models, its theory is simple and eliminates quantifiers up to some existential formulas - similar to ACFA. Moreover, it codes finite sets and allows a geometric elimination of imaginaries, but not always a weak elimination of imagina…

More on Galois cohomology, definability and differential algebraic groups

We make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired on Serre's algebraic twisting) to describe arbitrary fibres in cohomology sequences -- yielding a useful finiteness result on cohomology sets. Applied to the special case of differential fields and Kolchin's constrained cohomology, we prove that the first constrained cohomology set of a differen…

微分伽罗华理论中的参数化D- torsors

Parameterized D-torsors in differential Galois theory

本文研究了参数化D-torsors在微分伽罗华理论中的应用，提出了参数化版本的Kolchin定理，并探讨了广义强正常扩展与log-微分方程之间的关系。

p进闭域的伽罗华刻画——现代视角

The Galois characterisation of $p$-adically closed fields -- A modern perspective

本文通过新方法重新证明了p进闭域的伽罗华刻画，核心在于henselian性，并未依赖伽罗华上同调，同时联系了完美域和模型论转移技术。