We determine the dimensions of $\textrm{Ext}$ -groups between simple modules and dual generalized Verma modules in singular blocks of parabolic versions of category $\mathcal{O}$ for complex semisimple Lie algebras and affine Kac-Moody algebras.

John Rognes developed a notion of Galois extension of commutative ring spectra, and this includes a criterion for identifying an extension as unramified. Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by topological André–Quillen homology. In the classical algebraic context, it is important to distinguish between tame and wild ramification. Noether’s theorem characterizes tame ramification in terms of a normal basis, and tame ramification can also be detected via the surjectivity of the trace map. For commutative ring spectra, we suggest to study the Tate construction as a suitable analog. It tells us at which integral primes there is tame or wild ramification, and we determine its homotopy type in examples in the context of topological K-theory and topological modular forms.

Let $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;\;\sigma,\delta ]$ a skew polynomial ring.

Using skew polynomials $f\in R$ , we construct division algebras and maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include Jha Johnson semifields, and the classes of classical and twisted Gabidulin codes constructed by Sheekey.

Let $C_c^{*}(\mathbb{N}^{2})$ be the universal $C^{*}$ -algebra generated by a semigroup of isometries $\{v_{(m,n)}\,:\, m,n \in \mathbb{N}\}$ whose range projections commute. We analyse the structure of KMS states on $C_{c}^{*}(\mathbb{N}^2)$ for the time evolution determined by a homomorphism $c\,:\,\mathbb{Z}^{2} \to \mathbb{R}$ . In contrast to the reduced version $C_{red}^{*}(\mathbb{N}^{2})$ , we show that the set of KMS states on $C_{c}^{*}(\mathbb{N}^{2})$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.

We present a combinatorial proof for the existence of the sign-refined grid homology in lens spaces and a self-contained proof that $\partial _{\mathbb{Z}}^2 = 0$ . We also present a Sage programme that computes $\widehat{\mathrm{GH}} (L(p,q),K;\mathbb{Z})$ and provide empirical evidence supporting the absence of torsion in these groups.

Using totally symmetric sets, Chudnovsky–Kordek–Li–Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups.

In this paper, we study the Hausdorff dimension of sets defined by almost convergent binary expansion sequences. More precisely, the Hausdorff dimension of the following set

\begin{align*} \bigg\{x\in[0,1)\;:\;\frac{1}{n}\sum_{k=a}^{a+n-1}x_{k}\longrightarrow\alpha\textrm{ uniformly in }a\in\mathbb{N}\textrm{ as }n\rightarrow\infty\bigg\} \end{align*}

$ \alpha\in[0,1] $

$ \alpha $

In this short note, we deal with complete noncompact expanding and steady Ricci solitons of dimension $n\geq 3.$ More precisely, under an integrability assumption, we obtain a characterization for the generalized cigar Ricci soliton and the Gaussian Ricci soliton.

The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;\,c_1,c_2)$ to $Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ .

The Horikawa index and the local signature are introduced for relatively minimal fibered surfaces whose general fiber is a non-hyperelliptic curve of genus 4 with unique trigonal structure.

Using Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups.

Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open.

Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

This paper is about a type of quantitative density of closed geodesics and orthogeodesics on complete finite-area hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic and the shortest doubly truncated orthogeodesic that are $\varepsilon$ -dense on a given compact set on the surface.

A superelliptic curve over a discrete valuation ring $\mathscr{O}$ of residual characteristic p is a curve given by an equation $\mathscr{C}\;:\; y^n=\,f(x)$ , with $\textrm{Disc}(\,f)\neq 0$ . The purpose of this article is to describe the Galois representation attached to such a curve under the hypothesis that f(x) has all its roots in the fraction field of $\mathscr{O}$ and that $p \nmid n$ . Our results are inspired on the algorithm given in Bouw and WewersGlasg (Math. J.59(1) (2017), 77–108.) but our description is given in terms of a cluster picture as defined in Dokchitser et al. (Algebraic curves and their applications, Contemporary Mathematics, vol. 724 (American Mathematical Society, Providence, RI, 2019), 73–135.).

We prove a double commutant theorem for separable subalgebras of a wide class of corona C*-algebras, largely resolving a problem posed by Pedersen in 1988. Double commutant theorems originated with von Neumann, whose seminal result evolved into an entire field now called von Neumann algebra theory. Voiculescu later proved a C*-algebraic double commutant theorem for subalgebras of the Calkin algebra. We prove a similar result for subalgebras of a much more general class of so-called corona C*-algebras.

The main result includes as special cases on the one hand, the Gerstenhaber–Rothaus theorem (1962) and its generalisation due to Nitsche and Thom (2022) and, on the other hand, the Brodskii–Howie–Short theorem (1980–1984) generalising Magnus’s Freiheitssatz (1930).

We classify all $\mathbb{Q}$ -factorial Fano intrinsic quadrics of dimension three and Picard number one having at most canonical singularities.

An associative ring R is called potent provided that for every $x\in R$ , there is an integer $n(x)>1$ such that $x^{n(x)}=x$ . A celebrated result of N. Jacobson is that every potent ring is commutative. In this note, we show that a ring R is potent if and only if every nonzero subring S of R contains a nonzero idempotent. We use this result to give a generalization of a recent result of Anderson and Danchev for reduced rings, which in turn generalizes Jacobson’s theorem.

In the present note, we establish a finiteness theorem for $L^p$ harmonic 1-forms on hypersurfaces with finite index, which is an extension of the result of Choi and Seo (J. Geom. Phys.129 (2018), 125–132).

A subset E of a metric space X is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some n, sending E to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $\{E_i\}_{i=0}^{N+1}$ such that $E_{i}/E_{i+1}\subset X/E_{i+1}$ is starlike-equivalent for each i and $E_{N+1}$ is a point. A decomposition $\mathcal{D}$ of a metric space X is said to be recursively starlike-equivalent, if there exists $N\geq 0$ such that each element of $\mathcal{D}$ is recursively starlike-equivalent of filtration length N. We prove that any null, recursively starlike-equivalent decomposition $\mathcal{D}$ of a compact metric space X shrinks, that is, the quotient map $X\to X/\mathcal{D}$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman–Starbird and Freedman and is applicable to the proof of Freedman’s celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological 4-manifolds, including the four-dimensional Poincaré conjecture.

Let K be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, \mu$ for the existence of a nonzero homomorphism $\Delta(\lambda) \to \Delta (\mu)$ of Weyl modules for the general linear group $GL_n(K)$ . Also, for each p we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.