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We survey recent work on the geometry and dynamics of transverse subgroups of semi-simple Lie groups.
We survey some recent developments at the interface of algebraic geometry, surface topology, and the theory of ordinary differential equations. Motivated by "non-abelian" analogues of standard conjectures on the cohomology of algebraic…
The paper gives a short account of the contents of "Regular Algebraic K-Theory For Groups" by the author and its connections with other homology and K-theories.
A description of group automorphisms of all two-dimensional algebras, considered up to isomorphism, over any basic field is provided.
This is a survey on appearances of reflection groups, real and complex, in algebraic geometry. We also include a brief introduction into the theory of reflection groups.
The aim of this paper is to present the main constructions of the substructures of an almost groupoid and to discuss their basic properties. The definitions and properties concerning these new algebraic constructions extend to almost…
Algebraic geometry for groups and Lie algebraic has been recently defined and studied by many authors on the purpose to study set defined by algebraic equations on abstract groups and Lie algebras. The purpose of this paper is to present a…
This survey is based on my talk at the conference `Classical algebraic geometry today' at the MSRI. Some new results on the action of symplectomorphisms on the Chow group are added.
This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with…
We consider endomorphism actions of arbitrary discrete semigroups on a connected metrizable topological group G. We give necessary and sufficient conditions for expansiveness of such actions when G is a Lie group or a compact…
In this monograph we lay the foundation for a theory of coarse groups and coarse actions. Coarse groups are group objects in the category of coarse spaces, and can be thought of as sets with operations that satisfy the group axioms "up to…
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 study the globalization of partial actions on sets and topological spaces and of partial coactions on algebras by applying the general theory of globalization for geometric partial comodules, as previously developed by the authors. We…
There has been a great deal of research on graphs defined on algebraic structures in the last two decades. In this paper we begin an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this…
We explore the connection between ring homomorphisms and semigroup homomorphisms on matrix algebras over rings or $C^*$-algebras.
This a slightly expended version of my habilitation thesis, which is an overview of my research activities during the last 4 years, written in a rather informal style.
This article explains basic constructions and results on group algebras and their cohomology, starting from the point of view of commutative algebra. It provides the background necessary for a novice in this subject to begin reading Dave…
In this paper we give an algebraic characterization of assemblies in terms of bands of groups. We also consider substructures and homomorphisms of assemblies. We give many examples and counterexamples.
We analyze the abstract representations of the groups of rational points of even-dimensional quasi-split special unitary groups associated with quadratic field extensions. We show that, under certain assumptions, such representations have a…
This paper is an extended version of four lectures at PIMS in Vancouver given June 27 - 30, 2016. The primary goal of these lectures was to publicize the author's recent efforts to extend to representations of linear algebraic groups the…