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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.
Uniformly finite homology is a coarse invariant for metric spaces; in particular, it is a quasi-isometry invariant for finitely generated groups. In this article, we study uniformly finite homology of finitely generated amenable groups and…
A generalization of the notion of an $\infty$-category is presented, allowing for ($\infty$-)cat(egorie)s that may have non-invertible higher morphisms.
We classify the finite quasisimple groups whose commuting graph is perfect and we give a general structure theorem for finite groups whose commuting graph is perfect.
In this paper we introduce a new invariant (the distant degree) for difference field extensions of finite transcendence degree, and we explore some of its properties. We also discuss a generalisation of this invariant and of the limit…
We propose a general conjecture on decompositions of finite simple groups as products of conjugates of an arbitrary subset. We prove this conjecture for bounded subsets of arbitrary finite simple groups, and for large subsets of groups of…
This survey article is intended as an introduction to the recent categorical classification theorems of the three authors, restricting to the special case of the category of modules for a finite group.
A new definition for the notion of a (general) $\infty$-category is given.
We give a survey of recent developments in the investigation of the various local-global conjectures for representations of finite groups.
Various descending chains of subgroups of a finite permutation group can be used to define a sequence of `basic' permutation groups that are analogues of composition factors for abstract finite groups. Primitive groups have been the…
We give new bounds and asymptotic estimates on the largest Kronecker and induced multiplicities of finite groups. The results apply to large simple groups of Lie type and other groups with few conjugacy classes.
It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can…
We establish the ultra-violet finiteness of various classes of noncommutative gauge theories.
This paper includes new bounds concepting the vanishing generalized weighted Morrey space. In this sense, it is outlined improved bounds about the a class of fractional type rough higher order commutators on vanishing generalized weighted…
A complete classification of finitely generated involutive commutative two-valued groups is obtained. Three series of such two-valued groups are constructed: principal, unipotent and special, and it is shown that any finitely generated…
This expository article revolves around the question to find short presentations of finite simple groups. This subject is one of the most active research areas of group theory in recent times. We bring together several known results on…
The present note surveys my research related to generalizing notions of abelian group theory to non-commutative case and applying them particularly to investigate fundamental groups.
The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent…
In this note, we describe first the structure of minimal non-Iwasawa finite groups. Then we determine the minimal non-Iwasawa finite groups which are modular. Also, we find connections between minimal non-Iwasawa finite groups and the…
A survey of problems, conjectures, and theorems about quasi-isometric classification and rigidity for finitely generated solvable groups.