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We characterize group representations that factor through monomial representations, respectively, block-triangular representations with monomial diagonal blocks, by arithmetic properties. Similar results are obtained for semigroup…
In this article we introduce the notion of weak identities in a group and study their properties. We show that weak identities have some similar properties to ordinary ones. We use this notion to prove that any finitely generated solvable…
Controllable generation is one of the key requirements for successful adoption of deep generative models in real-world applications, but it still remains as a great challenge. In particular, the compositional ability to generate novel…
We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower…
Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph…
For a finite group G, we study the higher commuting probabilities, namely the probabilities that r randomly chosen elements of G commute pairwise, together with the corresponding numbers of simultaneous conjugacy classes of commuting…
In a seminal paper in 2009, Borcea, Br\"and\'en, and Liggett described the connection between probability distributions and the geometry of their generating polynomials. Namely, they characterized that stable generating polynomials…
We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a…
By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group has a derangement (an element with no fixed points). The existence of derangements with additional properties has attracted much attention,…
We construct finitely generated simple torsion-free groups with strong homological control. Our main result is that every subset of $\mathbb{N} \cup \{\infty\}$, with some obvious exceptions, can be realized as the set of dimensions of…
We introduce the bounded packing property for a subgroup of a countable discrete group G. This property gives a finite upper bound on the number of left cosets of the subgroup that are pairwise close in G. We establish basic properties of…
Generative Adversarial Networks (GANs) advance face synthesis through learning the underlying distribution of observed data. Despite the high-quality generated faces, some minority groups can be rarely generated from the trained models due…
This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames $\mathcal{F} = \{f_i\}_{i \in I}$ and $\mathcal{E} =…
We introduce a new invariant of finitely generated groups, the ambiguity function, and prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the…
Let $\mathcal{P}$ be the set of points of a finite-dimensional projective space over a local field $F$, endowed with the topology $\tau$ naturally induced from the canonical topology of $F$. Intuitively, continuous incidence abelian group…
In this work, we show how to parameterize a density matrix that has an arbitrary symmetry, knowing the generators of the Lie algebra (if the symmetry group is a connected Lie group) or the generators of its underlying group (in case it is…
We consider associative algebras with involution graded by a finite abelian group G over a field of characteristic zero. Suppose that the involution is compatible with the grading. We represent conditions permitting PI-representability of…
This paper investigates the maximal subgroups of a free projection-generated regular $*$-semigroup $PG(P)$ over a projection algebra $P$, and their relationship to the maximal subgroups of the free idempotent-generated semigroup $IG(E)$…
In this article we generalize a theorem of Benson for generalized quadrangles to strongly regular graphs and directed strongly regular graphs. The main result provides numerical restrictions on the number of fixed vertices and the number of…
This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of…