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The class-breadth conjecture of Leedham-Green, Neumann and Wiegold states that for each $p$-group, $cl(G)\leq b(G) + 1$, where $cl(G)$, $b(G)$ denote the nilpotency class and the breadth of $G$. While several counter-examples to this…

Let $k$ be a differential field of characteristic zero with an algebraically closed field of constants. In this article, we provide a classification of first order differential equations over $k$ and study the algebraic dependence of…

For a commutative Noetherian ring $R$ with finite Krull dimension, we study the nullity classes in $D^c_{fg}(R)$, the full triangulated subcategory $D^c_{fg}(R)$ of the derived category $D(R)$ consisting of objects which can be represented…

We calculate the Lusternik-Schnirelmann category of the k-th ordered configuration spaces F(R^n,k) of R^n and give bounds for the category of the corresponding unordered configuration spaces B(R^n,k) and the sectional category of the…

We develop the local-global theory of blocks for profinite groups. Given a field $k$ of characteristic $p$ and a profinite group $G$, one may express the completed group algebra $k[[G]]$ as a product $\prod_{i\in I}B_i$ of closed…

In a recent paper Kr\'olak and Beem have shown differentiability of Cauchy horizons at all points of multiplicity one. In this note we give a simpler proof of this result.

Every directed graph defines a Hilbert space and a family of weighted shifts that act on the space. We identify a natural notion of periodicity for such shifts and study their C*-algebras. We prove the algebras generated by all shifts of a…

In this paper, we apply the machinery developed in arXiv:2401.06641(2) to study the behavior of computable categoricity relativized to non-c.e. degrees. In particular, we show that we can build a computable structure which is not computably…

In this note we give a new existence proof for the universal extension classes for $GL_2$ previously constructed by Friedlander and Suslin via the theory of strict polynomial functors. The key tool in our approach is a calculation of Parker…

Spielberg's construction of C*-algebras from left cancellative small categories is a common generalization for most C*-algebras one would consider to come from ``combinatorial data,'' including graph and $k$-graph C*-algebras, Li's…

We study the unitarizability of premodular categories constructed from representations of quantum group at roots of unity. We introduce \emph{Grothendieck unitarizability} as a natural generalization of unitarizability to any class of…

A notion of exotic (ordered) configuration spaces of points on a space $X$ was suggested by Yu.~Baryshnikov. He gave equations for the (exponential) generating series of the Euler characteristics of these spaces. Here we consider un-ordered…

We characterize nonstandard models of ZF (of arbitrary cardinality) that can be expanded to Goedel-Bernays class theory plus $\Delta^1_1$-Comprehension. We also characterize countable nonstandard models of ZFC that can be expanded to…

Let $\mathrm{k}=\mathbb{Q}\left(\sqrt[3]{p},\zeta_3\right)$, where $p$ is a prime number such that $p \equiv 1 \pmod 9$, and let $C_{\mathrm{k},3}$ be the $3$-component of the class group of $\mathrm{k}$. In \cite{GERTH3}, Frank Gerth III…

We obtain the basic results concerning the problem of constructing operator realizations of the formal differential expression $\nabla \cdot a \cdot \nabla - b \cdot \nabla$ with measurable matrix $a$ and vector field $b$ having…

We study how to generate new Lie algebras $\mathcal{G}(N_0,..., N_p,...,N_n)$ from a given one $\mathcal{G}$. The (order by order) method consists in expanding its Maurer-Cartan one-forms in powers of a real parameter $\lambda$ which…

We classify differentiable structures on a line $\mathbb{L}$ with two origins being a non-Hausdorff but $T_1$ one-dimensional manifold obtained by ``doubling'' $0$. For $k\in\mathbb{N}\cup\{\infty\}$ let $H$ be the group of homeomorphisms…

We prove several new results on the structure of the subgroup generated by a small doubling subset of an ordered group, abelian or not. We obtain precise results generalizing Freiman's 3k-3 and 3k-2 theorems in the integers and several…

A characterization of the scalar type spectral generators of Beurling type Carleman ultradifferentiable $C_0$-semigroups is established, the important case of the Gevrey ultradifferentiability is considered in detail, the implementation of…

In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, we consider two functors from the category of Leibniz algebras to that of differential graded Lie algebras and show that they naturally give rise to…