Related papers: The relativized Lascar groups, type-amalgamation, …

We consider some examples of superintegrable system which were recently isolated through a differential Galois group analysis. The identity of these systems is clarified and the corresponding Poisson algebras derived.

We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.

In this note we classify simply connected rationally elliptic compact toric orbifolds up to algebraic isomorphism.

We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.

We establish the spectral gap property for dense subgroups generated by algebraic elements in any compact simple Lie group, generalizing earlier results of Bourgain and Gamburd for unitary groups.

We reduce the classification of finite subgroups in compact Lie groups to that of quasi-simple ones, prove the number of conjugacy classes is finite and each cojugacy class is Zariski closed in mapping space, and classify "strongly…

This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on…

We introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This allows us to compute all torsors for large classes of groups. We also develop some tools for…

We investigate properties of closed approximate subgroups of locally compact groups, with a particular interest for approximate lattices i.e. those approximate subgroups that are discrete and have finite co-volume. We prove an approximate…

For a stratified group $G$, we construct a class of polarised Lie groups, which we call modifications of $G$, that are locally contactomorphic to it. Vice versa, we show that if a polarised group is locally contactomorphic to a stratified…

A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and…

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups and we use structure theorems for these groups to…

Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained.

We explain how categories, and groupoids, can be seen as models for a Lawvere ${\mathfrak Gr}$-theory, where ${\mathfrak Gr}$ is the category of graphs, and show that for Lawvere ${\mathfrak Gr}$-theories finitely presentable models are…

This expository article introduces the topic of roots in a compact Lie group. Compared to the many other treatments of this standard topic, I intended for mine to be relatively elementary, example-driven, and free of unnecessary…

We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a…

We characterize relatively norm compact sets in the regular $C^*$-algebra of finitely generated Coxeter groups using a geometrically defined positive semigroup acting on the algebra.

The definition and basic properties of the Burnside ring of compact Lie groups are presented, with emphasis on the analogy with the construction of the Burnside ring of finite groups.

This paper provides a realization of all classical and most exceptional finite groups of Lie type as Galois groups over function fields over F_q and derives explicit additive polynomials for the extensions. Our unified approach is based on…

In this work, we introduce the concept of relative Lipschitz saturation, along with its key categorical and algebraic properties, and demonstrate how such a structure always gives rise to a radicial algebra.