Related papers: Recent developments in the theory of linear algebr…

This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…

This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has received relatively little attention, but…

This survey describes some recent work, by the authors and others, on the existence of algebraic fibrations of group extensions, as well as the finiteness properties of their algebraic fibers, in the realm of both abstract and pro-$p$…

We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields. In fact, we prove all of these conjectures for algebraic tori as well as in some other situations. This work relies in an…

Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds…

We first provide an overview of several results dealing with the genus of a division algebra and highlight the role of ramification in its analysis. We then give a survey of recent developments on the genus problem for simple algebraic…

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 investigate linearity of amalgams of subgroups of algebraic groups along intersections with algebraic subgroups. In the process, we establish linearity of certain "doubles" of linear groups, and obtain new examples of finitely generated…

Arithmetical properties of a finite group are properties of the group which are defined by its arithmetical parameters such as the order of the group, the element orders and so on. In this paper, we discuss a number of results on…

We obtain a lifting property for finite quotients of algebraic groups, and applications to the structure of these groups.

We report on recent progress concerning the relationship that exists between the algebraic structure of a finite group and certain features of its class-size prime graph.

This is an expository article on recent developments in the theory of group relaxations in integer programming from an algebraic perspective.

This is a brief introduction to the study of growth in groups of Lie type, with $SL_2(\mathbb{F}_q)$ and some of its subgroups as the key examples. They are an edited version of the notes I distributed at the Arizona Winter School in 2016.…

This paper presents a survey of the results and ideas behind the classification of the fine gradings, up to equivalence, on the simple finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It provides an…

We discuss a selection of recent developments in arithmetic combinatorics having to do with ``approximate algebraic structure'' together with some of their applications.

New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> makes an important such contribution.

We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…

We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the…

We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the…

This short survey contains some recent developments of the algebraic theory of racks and quandles. We report on some elements of representation theory of quandles and ring theoretic approach to quandles.