Related papers: Horn's problem in PU$(n,1)$

A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We prove that the multiple conjugacy problem is solvable between two n-tuples A and B of…

Let $A \subseteq E$ be a given extension of Hopf (respectively Lie) algebras. We answer the \emph{classifying complements problem} (CCP) which consists of describing and classifying all complements of $A$ in $E$. If $H$ is a given…

Let the complex reflection group $G(m,p,n)$ act on the unit polydisc $\mathbb D^n$ in $\mathbb C^n.$ A $\boldsymbol\Theta_n$-contraction is a commuting tuple of operators on a Hilbert space having…

Horn's problem asks for the conditions on sets of integers mu, nu and lambda that ensure the existence of Hermitian operators A, B and A+B with spectra mu, nu and lambda, respectively. It has been shown that this problem is equivalent to…

Horn's conjecture, which given the spectra of two Hermitian matrices describes the possible spectra of the sum, was recently settled in the affirmative. In this survey we discuss one of the many steps in this, which required us to introduce…

We revisit our construction of the Thompson groups from the polycyclic inverse monoids in the light of new research. Specifically, we prove that the Thompson group $G_{n,1}$ is the group of units of a Boolean inverse monoid $C_{n}$ called…

Let $n=2g+2$ be a positive even integer, $f(x)$ a degree $n$ complex polynomial without multiple roots and $C_f: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over the field $\C$ of complex numbers. Let a $(g-1)$-dimensional…

Let K be a fixed number field and G its absolute Galois group. We give a bound C(K), depending only on the degree, the class number and the discriminant of K, such that for any elliptic curve E defined over K and any prime number p strictly…

We present a new algorithm that, given two matrices in $GL(n,Q)$, decides if they are conjugate in $GL(n,Z)$ and, if so, determines a conjugating matrix. We also give an algorithm to construct a generating set for the centraliser in…

The volume and the number of lattice points of the permutohedron P_n are given by certain multivariate polynomials that have remarkable combinatorial properties. We give several different formulas for these polynomials. We also study a more…

We study the space $Q_n$ of all configurations of $n$ ordered points on the circle such that no three points coincide, and in which one of the points (say, the last one) is fixed. We compute its fundamental group for $n<6$ and describe its…

A generalization of the congruent number problem is to find positive integers $n$ that appear as the areas of Heron triangles. Selmer group of a congruent number elliptic curve has been studied quite extensively. Here, we look into the…

An element $\phi$ of the outer automorphism group $\Out(\f)$ of the rank $n$ free group $F_n$ is {\it polynomially growing} if the word lengths of conjugacy classes in $\f$ grow at most polynomially under iteration by $\phi$. It is {\it…

We study the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $GF(q)$ into orbits of the group $G=PGL(2,q)$ of linear symmetries of the twisted cubic $C$. A generic line neither intersects $C$ nor…

If $u$ and $v$ are two conjugate elements of a hyperbolic group then the length of a shortest conjugating element for $u$ and $v$ can be bounded by a linear function of the sum of their lengths, as was proved by Lysenok. Bridson and…

Let $H$ be a multiplicatively written monoid with identity $1_H$ and let $\mathcal{P}_{\text{fin},1}(H)$ denote the reduced finitary power monoid of $H$, that is, the monoid consisting of all finite subsets of $H$ containing $1_H$ with set…

In the framework of the spatial circular Hill three-body problem we illustrate the application of symplectic invariants to analyze the network structure of symmetric periodic orbit families. The extensive collection of families within this…

We prove that fundamental groups of orientable (geometrizable) 3-manifolds have a solvable conjugacy problem.

A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ for compact Lie groups $G$ and $H$ with $H$ 1-truncated,…

This paper gives a proof that the fundamental group of a class of closed orientable 3-manifolds constructed from three injective handlebodies has a solvable word problem. This is done by giving an algorithm to decide if a closed curve in…