Related papers: Combination Theorems in Groups, Geometry and Dynam…
Using the combinatorial properties of subsets of integers, a classification of metric dynamical systems was given in [V. Bergelson and T. Downarowicz, Large sets of integers and hierarchy of mixing properties of measure-preserving systems,…
Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective…
We discuss and compare several geometric structures which imply an upper bound to the acceleration of a particle measured in its rest system. While all of them have the same implications on the motion of a point particle, they differ in…
We formulate a number of new results in Algebraic Geometry and outline their derivation from Theorem 2.12 which belongs to Algebraic Combinatorics.
Embedding the data in hyperbolic spaces can preserve complex relationships in very few dimensions, thus enabling compact models and improving efficiency of machine learning (ML) algorithms. The underlying idea is that hyperbolic…
In 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a…
Coordination geometries describe how the neighbours of a central particle are arranged around it. Such geometries can be thought to lie in an abstract topological space; a model of this space could provide a mathematical basis for…
We remark the importance of adding suitable pre-geometric content to tensor models, obtaining what has recently been called tensorial group field theories, to have a formalism that could describe the structure and dynamics of quantum…
I present in this paper some tools in Symplectic and Poisson Geometry in view of their applications in Geometric mechanics and Mathematical Physics. After a short discussion of the Lagrangian and Hamiltonian formalisms, including the use of…
This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings,…
In this thesis manuscript we explore different facets of random tensor models. These models have been introduced to mimic the incredible successes of random matrix models in physics, mathematics and combinatorics. After giving a very short…
The group of automorphisms of the geometry of an integrable system is considered. The geometrical structure used to obtain it is provided by a normal form representation of integrable systems that do not depend on any additional geometrical…
Parabolic cut pairs in the boundaries of relatively hyperbolic group are a new and previously unexplored phenomenon. In this paper, we give a way to create examples of relatively hyperbolic groups with parabolic cut pairs on their boundary…
These are a few historical remarks, addenda and references with comments on some topics discussed by Thurston in his notes ''The geometry and topology of three-manifolds''. The topics are mainly hyperbolic geometry, geometric structures,…
The purpose of this note is to give an exposition of some interesting combinatorics and convex geometry concepts that appear in algebraic geometry in relation to counting the number of solutions of a system of polynomial equations in…
We present a careful approximation of the geodesics in trees of hyperbolic or relatively hyperbolic groups. As an application we prove a combination theorem for finite graphs of relatively hyperbolic groups, with both Farb's and Gromov's…
We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connec-tion between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric…
This is a survey. The main subject of this survey is the homotopical or homological nature of certain structures which appear in classical problems about groups, Lie rings and group rings. It is well known that the (generalized) dimension…
We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into…
The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.