Résultats de recherche pour « Hyperbolic Geometry »

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure.

Cosmological $\ensuremath{\alpha}$ attractors give a natural explanation for the spectral index ${n}_{s}$ of inflation as measured by Planck while predicting a range for the tensor-to-scalar ratio $r$, consistent with all observations, to be measured more precisely in future B-mode experiments. We highlight the crucial role of the hyperbolic geometry of the Poincar\'e disk or half plane in the supergravity construction. These geometries are isometric under M\"obius transformations, which include the shift symmetry of the inflaton field. We introduce a new K\"ahler potential frame that explicitly preserves this symmetry, enabling the inflaton to be light. Moreover, we include higher-order curvature deformations, which can stabilize a direction orthogonal to the inflationary trajectory. We illustrate this new framework by stabilizing the single superfield $\ensuremath{\alpha}$ attractors.

This will be a description of a few highlights in the early history of non-euclidean geometry, and a few miscellaneous recent developments.An Appendix describes some explicit formulas concerning volume in hyperbolic 3-space.The mathematical literature on non-euclidean geometry begins in 1829 with publications by N. Lobachevsky in an obscure Russian journal.The infant subject grew very rapidly.Lobachevsky was a fanatically hard worker, who progressed quickly from student to professor to rector at his university of Kazan, on the Volga.Already in 1829, Lobachevsky showed that there is a natural unit of distance in non-euclidean geometry, which can be characterized as follows.In the right triangle of Figure 1 with fixed edge a, as the opposite vertex A moves infinitely far away, the angle 9 will increase to a limit 9 0 which is assumed to be strictly less than 7r/2.He showed that a = -log tan(0 o /2) if the unit of distance is suitably chosen.In particular, a « (TT/2) -0 O if a is very small.(In the interpretation introduced by Beltrami forty years later, this unit of distance is chosen so that curvature = -1.)

In the natural environment, the sense of smell, or olfaction, serves to detect toxins and judge nutritional content by taking advantage of the associations between compounds as they are created in biochemical reactions. This suggests that the nervous system can classify odors based on statistics of their co-occurrence within natural mixtures rather than from the chemical structures of the ligands themselves. We show that this statistical perspective makes it possible to map odors to points in a hyperbolic space. Hyperbolic coordinates have a long but often underappreciated history of relevance to biology. For example, these coordinates approximate the distance between species computed along dendrograms and, more generally, between points within hierarchical tree-like networks. We find that both natural odors and human perceptual descriptions of smells can be described using a three-dimensional hyperbolic space. This match in geometries can avoid distortions that would otherwise arise when mapping odors to perception.

1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the

Abstract Every compact surface admits a constant curvature metric, endowing it with a geometry. For surfaces of genus at least 2, this leads to hyperbolic geometry. The model space is the hyperbolic plane. We give a brief introduction to the beautiful world of hyperbolic geometry in dimension 2 starting from the hyperbolic plane and its isometries and then discussing hyperbolic surfaces. References for an extended treatment include, for example, Thurston (Collected works of William P. Thurston with commentary: IV. The geometry and topology of three-manifolds: with a preface by Steven P. Kerckhoff, 2022 ) or Benedetti and Petronio (Lectures on hyperbolic geometry, 1992); Marden (Hyperbolic manifolds. An introduction in 2 and 3 dimensions, 2016); Ramsay and Richtmyer (Introduction to hyperbolic geometry, 1995).