Related papers: What is a hierarchically hyperbolic space?

In this paper we continue the study of non-diagonalisable hyperbolic systems with variable multiplicity started by the authors in \cite{Garetto2018}. In the case of space dependent coefficients, we prove a representation formula for…

We provide examples of transitive partially hyperbolic dynamics (specific but paradigmatic examples of homoclinic classes) which blend different types of hyperbolicity in the one-dimensional center direction. These homoclinic classes have…

Hyperbolic embeddings are a class of representation learning methods that offer competitive performances when data can be abstracted as a tree-like graph. However, in practice, learning hyperbolic embeddings of hierarchical data is…

We discuss several ways of packing a hyperbolic surface with circles (of either varying radii or all being congruent) or horocycles, and note down some observations related to their symmetries (or the absence thereof).

We investigate unbounded domains in hierarchically hyperbolic groups and obtain constraints on the possible hierarchical structures. Using these insights, we characterise the structures of virtually abelian HHGs and show that the class of…

We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has…

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.

Label inventories for fine-grained entity typing have grown in size and complexity. Nonetheless, they exhibit a hierarchical structure. Hyperbolic spaces offer a mathematically appealing approach for learning hierarchical representations of…

In this work, we are concerned with hierarchically hyperbolic spaces and hierarchically hyperbolic groups. Our main result is a wide generalization of a combination theorem of Behrstock, Hagen, and Sisto. In particular, as a consequence, we…

Volume is a natural measure of complexity of a Riemannian manifold. In this survey, we discuss the results and conjectures concerning n-dimensional hyperbolic manifolds and orbifolds of small volume.

This paper collects some important formulas on hyperbolic volume. To determine concrete values of volume function is a very hard question requiring the knowledge of various methods. Our goal to give a non-elementary integral on the volume…

Through semiclassical methods the subject of quantum chaos motivates and depends on Hamiltonian chaos research. Presented here is a selection of Hamiltonian chaos topics that in this way get directly related to any of a variety of quantum…

In this article we introduce a new geometric object called hyperbolic Pascal simplex. This new object is presented by the regular hypercube mosaic in the 4-dimensional hyperbolic space. The definition of the hyperbolic Pascal simplex, whose…

This is the first of the two articles where we determine the higher smooth surgery structure sets of complex projective spaces (up to some extension problems) and the forgetful map to their topological versions in low dimensions. In this…

We show that many important natural science models in their mathematical formulation can be reduced to non-strictly hyperbolic systems of the same kind. This allows the same methods to be applied to them so that some essential results…

In this article we give evaluations of certain series of hyperbolic functions, using Jacobi elliptic functions theory. We also define some new functions that enable us to give characterization of not solvable class of series.

Aim of this paper is to review some basic ideas and recent developments in the theory of strictly hyperbolic systems of conservation laws in one space dimension. The main focus will be on the uniqueness and stability of entropy weak…

We discuss asymptotically hyperbolic manifold with a noncompact boundary which is close to a horosphere in a certain sense. The model case is a horoball or the complement of a horoball in standard hyperbolic space. We show some geometric…

We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and…

This is brief and hopefully friendly, with basic notions, a few different perspectives, and references with more information in various directions.