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An \emph{s-graph} is a graph with two kinds of edges: \emph{subdivisible} edges and \emph{real} edges. A \emph{realisation} of an s-graph $B$ is any graph obtained by subdividing subdivisible edges of $B$ into paths of arbitrary length (at…

For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple restricted Lie algebras of types W(m;1) and S(m;1) (m>=2), in terms of numerical and group-theoretical invariants. Our main tool is automorphism…

Let $S_{g,n}$ be an oriented surface of genus $g$ with $n$ punctures, where $2g-2+n>0$ and $n>0$. Any ideal triangulation of $S_{g,n}$ induces a global parametrization of the Teichm\"uller space $\mathcal{T}_{g,n}$ called the shearing…

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $Mod(C)$, from a skeletally small preadditive category $C$ to the category of abelian groups. We introduced the notion of a a…

We study the moduli space $\mathcal{AS}_{g}$ of Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of positive characteristic $p$. The moduli space is partitioned by irreducible strata, where each stratum…

Let $\Gamma_{g,1}^m$ be the mapping class group of the orientable surface $\Sigma_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let…

We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Gamma is any graph not containing a four-cycle without chords, then the group G(Gamma) does not contain four elements whose…

Let $N_g$ be a closed, connected, nonorientable surface of genus $g$. We prove that for $g \ge 13$, the mapping class group $\text{Mod}(N_g)$ can be generated by exactly two elements. This improves the previously known bound of $g \ge 19$.

A connected graph $G$ is of QE class if it admits a quadratic embedding in a Hilbert space, or equivalently if the distance matrix is conditionally negative definite, or equivalently if the quadratic embedding constant $\mathrm{QEC}(G)$ is…

In this paper we study the elementary theory of graph products of groups and show that under natural conditions on the vertex groups we can recover (the core of) the underlying graph and the associated vertex groups. More precisely, we…

We investigate which classes of infinite graphs have the Erd\H{o}s-P\'osa property (EPP). In addition to the usual EPP, we also consider the following infinite variant of the EPP: a class $\mathcal{G}$ of graphs has the $\kappa$-EPP, where…

We prove the following CR version of Artin's approximation theorem for holomorphic mappings between real-algebraic sets in complex space. Let $M\subset \C^N$ be a real-algebraic CR submanifold whose CR orbits are all of the same dimension.…

A groupoid G is called an AG-groupoid if it satisfies the left invertive law: (ab)c = (cb)a. An AG-group G, is an AG-groupoid with left identity e \in G (that is, ea = a for all a \in G) and for all a \in G there exists a' \in G such that…

An example of a finite dimensional factorizable ribbon Hopf C-algebra is given by a quotient H=u_q(g) of the quantized universal enveloping algebra U_q(g) at a root of unity q of odd degree. The mapping class group M_{g,1} of a surface of…

We consider two natural embeddings between Artin groups: the group G_{tilde{A}_{n-1}} of type tilde{A}_{n-1} embeds into the group G_{B_n} of type B_n; G_{B_n} in turn embeds into the classical braid group Br_{n+1}:=G_{A_n} of type A_n. The…

Let A be an algebra in a variety V. We study the modulization of a pointed A-overalgebra P, show that it is totally in any variety that P is totally in, and apply this theory to the construction of the enveloping ringoid Z[A,V].

Let $\mathbb{Z}_{p^s}$ be the residue class ring of integers modulo $p^s$, where $p$ is a prime number and $s$ is a positive integer. Using matrix representation and the inner rank of a matrix, we study the intersection, join, dimension…

We consider a priori estimates of Weyl's embedding problem of $(\mathbb{S}^2, g)$ in general $3$-dimensional Riemannian manifold $(N^3, \bar g)$. We establish interior $C^2$ estimate under natural geometric assumption. Together with a…

Let $\mathfrak{G}$ be the subgroup $\mathfrak{S}_{n-q} \times \mathfrak{S}_{q}$ of the $n$-th symmetric group $\mathfrak{S}_{n}$ for $n-q \geq q$. In this paper, we study the $\mathfrak{G}$-invariant part of the rational cohomology group of…

Deformation spaces Hom($\pi$,G)/G of representations of the fundamental group $\pi$ of a surface $\Sigma$ in a Lie group $G$ admit natural actions of the mapping class group $Mod_\Sigma$, preserving a Poisson structure. When $G$ is compact,…