Related papers: A lower bound on the essential dimension of a conn…

We consider $G$, a linear group defined over $k$, an algebraically closed field. By considering $k$ as an embedded residue field of an algebraically closed valued field $K$, we can associate to it a compact $G$-space $S^\mu_G(k)$,…

We consider a twist $E$ over an \'etale groupoid $G$. When $G$ is principal, we prove that the nuclear dimension of the reduced twisted groupoid $\mathrm{C}^*$-algebra is bounded by a number depending on the dynamic asymptotic dimension of…

We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we…

Let $n,p,r$ be positive integers with $n \geq p\geq r$. A rank-$\overline{r}$ subset of $n$ by $p$ matrices (with entries in a field) is a subset in which every matrix has rank less than or equal to $r$. A classical theorem of Flanders…

The Gordon--Rodriguez-Villegas theorem says that, in a finite group, the number of solutions to a system of coefficient-free equations is divisible by the order of the group if the rank of the matrix composed of the exponent sums of $j$-th…

Let $G$ be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank $\geq 2$, and let $R$ be a commutative ring. We analyze the linear representations $\rho \colon G(R)^+ \to GL_n (K)$ over an…

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. In a recent work, we have determined the maximal dimension for a linear subspace of $n$ by $n$ symmetric matrices with rank less than or equal to…

Let $k$ be an algebraically closed field of characteristic $p \geq 0$, let $G$ be a simple simply connected classical linear algebraic group of rank $\ell$ and let $T$ be a maximal torus in $G$ with rational character group $X(T)$. For a…

We introduce the nuclear dimension of a C*-algebra; this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank. Our notion behaves well with respect to inductive…

We investigate when discrete, amenable groups have $C^*$-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse in an open problem. We show that if $C^*(G)$ has real rank zero, then…

Let G be a connected simple algebraic group over an algebraically closed field k of characteristic p > 0, and g := Lie(G). We additionally assume that G is standard and is of type An. Motivated by the investigation of the geometric…

Let $k$ be an algebraically closed field of characteristic 0, $Y=k^{r}\times {(k^{\times})}^{s}$ and let $G$ be an algebraic torus acting diagonally on the ring of differential operators $\cD (Y)^G$. We give necessary and sufficient…

Mean dimension for AH-algebras is introduced. It is shown that if a simple unital AH-algebra with diagonal maps has mean dimension zero, then it has strict comparison on positive elements. In particular, the strict order on projections is…

We show that unital simple C*-algebras with tracial topological rank zero which are locally approximated by subhomogeneous C^-algebras can be classified by their ordered $K$-theory. We apply this classification result to show that certain…

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric…

A ring $R$ with center $C$ is said to be \textit{centrally essential} if the module $R_C$ is an essential extension of the module $C_C$. In the paper, we study groups whose group algebras over fields are centrally essential rings. We focus…

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup…

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $p\geq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and…

We explore various limit constructions for C*-algebras, such as composition series and inverse limits, in relation to the notions of real rank, stable rank, and extremal richness. We also consider extensions and pullbacks. We identify some…

An unrefinable chain of a finite group $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal subgroup of $G_{i-1}$. The length (respectively, depth) of $G$ is the maximal (respectively, minimal)…