矩阵环的模型论视角

On relationships between algebraic properties of groups and rings in some model-theoretic contexts

We study relationships between certain algebraic properties of groups and rings definable in a first order structure or $*$-closed in a compact $G$-space. As a consequence, we obtain a few structural results about $ω$-categorical rings as well as about small, $nm$-stable compact $G$-rings, and we also obtain surprising relationships between some conjectures concerning small profinite groups.

Remarks on expansions of the real field: tameness, Hardy fields and smooth rings

In the talk \cite{vandendries-matthias-} presented at Logic and Foundations section of ICM-2018, Rio de Janeiro, the authors analyze, under a model-theoretic perspective, three ways to enrich the real continuum by infinitesimal and infinite quantities. In the present work, we present a first model-theoretic connection of another (but related to the previous one) triple of structures: o-minimal structures, Hardy fields and smooth rings.

Bohr compactifications of groups and rings

We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group $UT_3(Z)$, the continuous Heisenberg group $UT_3(R)$, and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.

Fraisse's construction from a topos-theoretic perspective

We present a topos-theoretic interpretation of (a categorical generalization of) Fraisse's construction in model theory, with applications to countably categorical theories.

半单环上模的模型论 $K_1$：（弱）Morita 不变性

Model-theoretic $K_1$ for modules over semisimple rings: (weak) Morita invariance

本文计算了除环上矩阵环模的模型论 $K_1$，证明了除环的（弱）Morita 不变性，并进一步计算了半单环上模的 $K_1$，同时给出代数 $K_1$ 到模型论 $K_1$ 的嵌入结果。

算术层级：一个可实现性理论的视角

The Arithmetical Hierarchy: A Realizability-Theoretic Perspective

本文从可实现性理论视角研究算术层级，发现将可实现性解释与多一归约结合后，可对自然算术问题实现非平凡分类。