Related papers: The congruence subgroup problem

In this note, we give a brief survey on some recent developments of biharmonic submanifolds. After reviewing some recent progress on Chen's biharmonic conjecture, the Generalized Chen's conjecture on biharmonic submanifolds of…

It is an interesting and open problem to trace the origin of the pseudospin symmetry in nuclear single-particle spectra and its symmetry breaking mechanism in actual nuclei. In this report, we mainly focus on our recent progress on this…

In order to make the fundamental group, one of the most well known invariants in algebraic topology, more useful and powerful some researchers have introduced and studied various topologies on the fundamental group from the beginning of the…

An $integral$ of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. This paper continues the investigation on integrals of groups started in the work arXiv:1803.10179. We study: (1) A sufficient condition for a bound…

An integral of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. In this paper, we prove that the integrability of a finite group is a decidable problem.

Given a combinatorial structure, a ``twin'' is a pair of disjoint substructures which are isomorphic (or look the same in some sense). In recent years, there have been many problems about finding large twins in various combinatorial…

In this paper we address the issue of existence of cusp forms for almost simple Lie groups using the approach of the second author combined with local information on supercuspidal representations for $p$-adic groups known by the first…

We establish a correspondence between generalized quiver gauge theories in four dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are…

This article introduces patterns of ideals of numerical semigroups, thereby unifying previous definitions of patterns of numerical semigroups. Several results of general interest are proved. More precisely, this article presents results on…

In this article we study the combinatorics of congruence subgroups of the modular group. We consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes),…

Let $A$ be the ring of elements in an algebraic function field $K$ over a finite field $F_q$ which are integral outside a fixed place $\infty$. In an earlier paper we have shown that the Drinfeld modular group $G=GL_2(A)$ has automorphisms…

For a given inverse semigroup, one can associate an \'etale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated \'etale groupoids. In this paper, we focus on…

In the paper we study inverse semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and $\bar{\mathscr{B}}\,^+(G)$ which are generated by partial monotone injective translations of a positive cone of a linearly ordered…

New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> makes an important such contribution.

We derive an explicit isomorphism between the Hilbert modular group and certain congruence subgroups on the one hand and particular subgroups of the special orthogonal group $SO(2, 2)$ on the other hand. The proof is based on an application…

This is a brief survey of recent results related to austere submanifolds, mainly based on the papers [24,25].

A long-standing open question in Integer Programming is whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are congruency-constrained integer programs…

Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to…

The structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group is related to a notion of Hilbert modules endowed with inner products taking values in spaces of unbounded operators. A…

We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…