Related papers: Reciprocity obstructions in semigroup orbits in SL…

We study group congruences on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ and its homomorphic retracts in the case when an ${\omega}$-closed family $\mathscr{F}$ which consists of inductive non-empty subsets of $\omega$. It is…

Given an abelian, CM extension K of any totally real number field k, we consider two conjectures `of Stark type'. The `Integrality Conjecture' concerns the image of a p-adic map `\mathfrak{s}_{K/k,S}' determined by the minus-part of the…

We consider box spaces of finitely generated, residually finite groups $G$, and try to distinguish them up to coarse equivalence. We show that, for $n\geq 2$, the group $SL_n(\mathbb{Z})$ has a continuum of box spaces which are pairwise…

Given a point and an expanding map on the unit interval, we consider the set of points for which the forward orbit under this map is bounded away from the given point. For maps like multiplication by an integer modulo 1, such sets have full…

We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated `Latin Square Conjecture' due to Alon-Tarsi or more…

The Schr\"odinger-Robertson inequality for relativistic position and momentum operators X^\mu, P_\nu, \mu, \nu = 0,1,2,3, is interpreted in terms of Born reciprocity and `non-commutative' relativistic phase space geometry. For states which…

A permutation $\pi$ contains a pattern $\sigma$ if and only if there is a subsequence in $\pi$ with its letters are in the same relative order as those in $\sigma$. Partially ordered patterns (POPs) provide a convenient way to denote…

This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts…

We investigate semigroups $S$ which have the property that every subsemigroup of $S\times S$ which contains the diagonal $\{ (s,s)\colon s\in S\}$ is necessarily a congruence on $S$. We call such $S$ a DSC semigroup. It is well known that…

The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose $G$ is a simple algebraic group over the rational numbers satisfying both strong approximation, and the…

We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime $p$, extending previous work of Shparlinski (2017).

Let $G$ be a nontrivial permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is…

It is a folk conjecture that for alpha > 1/2 there is no alpha-Hoelder surface in the subRiemannian Heisenberg group. Namely, it is expected that there is no embedding from an open subset of R^2 into the Heisenberg group that is Hoelder…

We obtain a power saving in the error term for a semigroup congruence lattice point count related to continued fractions. This is done by adapting arguments from recent work of Oh and Winter (2014) that give uniform bounds for certain…

We present a variant of the Peskine--Szpiro Acyclicity Lemma, and hence a way to certify exactness of a complex of finite modules over a large class of (possibly) noncommutative rings. Specifically, over the class of Auslander regular…

It is a longstanding conjecture that for a finite group $G$, the exponent of the second homology group $H_2(G, \mathbb{Z})$ divides the exponent of $G$. In this paper, we prove this conjecture for $p$-groups of class at most $p$, finite…

Using recent results of the second author which explicitly identify the "$(1,2,1,2)$-avoiding" $GL(p,\mathbb{C}) \times GL(q,\mathbb{C})$-orbit closures on the flag manifold $GL(p+q,\mathbb{C})/B$ as certain Richardson varieties, we give…

In this paper, the two settings we are concerned with are $\Gamma < \operatorname{SO}(n, 1)$ a Zariski dense Schottky semigroup and $\Gamma < \operatorname{SL}_2(\mathbb C)$ a Zariski dense continued fractions semigroup. In both settings,…

We extend the Gutzwiller trace formula to systems of noninteracting identical particles. The standard relation for isolated orbits does not apply since the energy of each particle is separately conserved causing the periodic orbits to occur…

Katugampola's 2015 study of generalized fractional differential operators produced triangular arrays of integer coefficients indexed by a fractional order r and by dimensions n and k, but no combinatorial interpretation has been established…