Let $p \;:\; Y \to X$ be a finite, regular cover of finite graphs with associated deck group $G$, and consider the first homology $H_1(Y;\;{\mathbb{C}})$ of the cover as a $G$-representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group $G$ on the one hand and topological properties of homology classes in $H_1(Y;\;{\mathbb{C}})$ on the other hand. We do so by studying certain subrepresentations in the $G$-representation $H_1(Y;\;{\mathbb{C}})$.

The homology class of a lift of a primitive element in $\pi _1(X)$ spans an induced subrepresentation in $H_1(Y;\;{\mathbb{C}})$, and we show that this property is never sufficient to characterize such homology classes if $G$ is Abelian. We study $H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \leq H_1(Y;\;{\mathbb{C}})$—the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in $\pi _1(X)$. Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with $H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \neq \ker\!(p_*)$.

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.

Let $C_{\||.\||}$ be an ideal of compact operators with symmetric norm $\||.\||$ . In this paper, we extend the van Hemmen–Ando norm inequality for arbitrary bounded operators as follows: if f is an operator monotone function on $[0,\infty)$ and S and T are bounded operators in $\mathbb{B}(\mathscr{H}\;\,)$ such that ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a=\{z\in \mathbb{C} \ | \ {\rm{re}}(z)\geq a\}$ , then

\begin{equation*}\||f(S)X-Xf(T)\|| \leq\;f'(a) \ \||SX-XT\||,\end{equation*}

$X\in C_{\||.\||}$

${\rm{sp}}(S), {\rm{sp}}(T) \subseteq \Gamma_a$

\begin{equation*}\||S^r X-XT^r\|| \leq r a^{r-1} \ \||SX-XT\||,\end{equation*}

$X\in C_{\||.\||}$

$0\leq r\leq 1$

We calculate the moments of the characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, at a position t in the bulk of the spectrum, as a series expansion in powers of t. We focus in particular on the Gaussian Unitary Ensemble. We employ a novel approach to calculate the coefficients in this series expansion of the moments, appropriately scaled. These coefficients are polynomials in N. They therefore grow as $N\to\infty$ , meaning that in this limit the radius of convergence of the series expansion tends to zero. This is related to oscillations as t varies that are increasingly rapid as N grows. We show that the $N\to\infty$ asymptotics of the moments can be derived from this expansion when $t=0$ . When $t\ne 0$ we observe a surprising cancellation when the expansion coefficients for N and $N+1$ are formally averaged: this procedure removes all of the N-dependent terms leading to values that coincide with those expected on the basis of previously established asymptotic formulae for the moments. We obtain as well formulae for the expectation values of products of the secular coefficients.

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.

We classify embeddings of the finite groups $A_4$ , $S_4$ and $A_5$ in the Lie group $G_2(\mathbb C)$ up to conjugation.

In this work, we study foliations of arbitrary codimension $\mathfrak{F}$ with integrable normal bundles on complete Riemannian manifolds. We obtain a necessary and sufficient condition for $\mathfrak{F}$ to be totally geodesic. For this, we introduce a special number $\mathfrak{G}_{\mathfrak{F}}^{\alpha}$ that measures when the foliation ceases to be totally geodesic. Furthermore, applying some maximum principle we deduce geometric properties for $\mathfrak{F}$ . We conclude with a geometrical version of Novikov’s theorem (Trans. Moscow Math. Soc. (1965), 268–304), for Riemannian compact manifolds of arbitrary dimension.

This paper is devoted to the study of generalized tilting theory of functor categories in different levels. First, we extend Miyashita’s proof (Math Z 193:113–146,1986) of the generalized Brenner–Butler theorem to arbitrary functor categories $\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C})$ with $\mathcal{C}$ an annuli variety. Second, a hereditary and complete cotorsion pair generated by a generalized tilting subcategory $\mathcal{T}$ of $\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C})$ is constructed. Some applications of these two results include the equivalence of Grothendieck groups $K_0(\mathcal{C})$ and $K_0(\mathcal{T})$, the existences of a new abelian model structure on the category of complexes $\mathop{\textrm{C}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C}))$, and a t-structure on the derived category $\mathop{\textrm{D}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C}))$.

We show via $\ell^2$ -homology that the rational homological dimension of a lattice in a product of simple simply connected Chevalley groups over global function fields is equal to the rational cohomological dimension and to the dimension of the associated Bruhat–Tits building.

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).

Erdős considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes p for which there is no $\mathbb{Q}_p$ -point on a random variety are Poisson distributed.

In this work, we study the Humbert-Edge curves of type 5, defined as a complete intersection of four diagonal quadrics in ${\mathbb{P}}^5$. We characterize them using Kummer surfaces, and using the geometry of these surfaces, we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we show how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-1}{2}$ and the moduli space of Humbert-Edge curves of type $n\geq 5$ where $n$ is an odd number.

We present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type $A_2$ with $q=-1$ , a.k.a. the positive part of the small quantum group $\mathfrak{u}^+_{\sqrt{-\text{1}}}(\mathfrak{sl}_3)$ . We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.

A. Mark and J. Paupert [Presentations for cusped arithmetic hyperbolic lattices, 2018, arXiv:1709.06691.] presented a method to compute a presentation for any cusped complex hyperbolic lattice. In this note, we will use their method to give a presentation for the Eisenstein-Picard modular group in three complex dimensions.

We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroups are infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics are stable when passing to commensurable groups, by understanding their twisted conjugacy growth. We also use these estimates to prove that, in certain cases, the conjugacy growth series cannot be a holonomic function.

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.

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 study Cayley graphs of abelian groups from the perspective of quantum symmetries. We develop a general strategy for determining the quantum automorphism groups of such graphs. Applying this procedure, we find the quantum symmetries of the halved cube graph, the folded cube graph, and the Hamming graphs.

We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields of arbitrary characteristic. Furthermore, from the same perspective, we study a family of cluster algebras which are not of finite type and which arise from a star shaped quiver.