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We define a family of groups that generalises Thompson's groups $T$ and $G$ and also those of Higman, Stein and Brin. For groups in this family we descrine centralisers of finite subgroups and show, that for a given finite subgroup $Q$,…

We consider and characterize classes of finite and countably categorical structures and their theories preserved under $E$-operators and $P$-operators. We describe $e$-spectra and families of finite cardinalities for structures belonging to…

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…

We develop cohomological and homological theories for a profinite group $G$ with coefficients in the Pontryagin dual categories of pro-discrete and ind-profinite $G$-modules, respectively. The standard results of group (co)homology hold for…

We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincar\'e inequality holds. We show that the constant rank condition is sufficient. The concept of the…

We prove special cases of a general conjecture: If an invertible field theory admits a projectively topological boundary theory, then it has finite order in the abelian group of invertible field theories. One can substitute `gapped' for…

A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…

In this paper we study the general group classification of systems of linear second-order ordinary differential equations inspired from earlier works and recent results on the group classification of such systems. Some interesting results…

We give a survey of recent developments in the investigation of the various local-global conjectures for representations of finite groups.

In this paper we prove some general theorems about representations of finite groups arising from the inner semidirect product of groups. We show how these results can be used for standard applications of group theory in quantum chemistry…

In this survey paper we discuss some recent results and related open questions in additive combinatorics, in particular, questions about sumsets in finite abelian groups.

In this paper, by using the Groebner-Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier…

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the $G$-comodules…

The complete group classification of a generalization of the Black-Scholes-Merton model is carried out by making use of the underlying equivalence and additional equivalence transformations. For each non linear case obtained through this…

This note presents Galois theory for finite fields. It was written as a handout for the MAT401 course ``Polynomial equations and fields'' taught at the University of Toronto in Spring 2026. We use without proofs some basic properties of…

We introduce the notion of a pro-fusion system on a pro-p group, which generalizes the notion of a fusion system on a finite p-group. We also prove a version of Alperin's Fusion Theorem for pro-fusion systems.

We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating…

In this paper we introduce and study the concept of normality degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be normal. Explicit formulas are obtained for some particular classes of…

We apply Kac's theory of elements of finite order (EFO) in Lie groups to the description of discrete gauge symmetries in various supersymmetric grand unified models. Taking into account the discrete anomaly cancellation conditions, we…

We establish the structure of finite groups with $\mathfrak{F}$-subnormal or self-normalizing primary cyclic subgroups in case $\mathfrak{F}$ is a subgroup-closed saturated superradical formation containing all nilpotent groups.