Geometry of Space

In recent years hyperbolic geometry has been the object and the preparation for extensive study that has produced important and often amazing results and also opened up new questions. The book concerns the geometry of manifolds and in particular hyperbolic manifolds; its aim is to provide an exposition of some fundamental results, and to be as far as possible self-contained, complete, detailed and unified. Since it starts from the basics and it reaches recent developments of the theory, the book is mainly addressed to graduate-level students approaching research, but it will also be a helpful and ready-to-use tool to the mature researcher. After collecting some classical material about the geometry of the hyperbolic space and the Teichmüller space, the book centers on the two fundamental results: Mostow's rigidity theorem (of which a complete proof is given following Gromov and Thurston) and Margulis' lemma. These results form the basis for the study of the space of the hyperbolic manifolds in all dimensions (Chabauty and geometric topology); a unified exposition is given of Wang's theorem and the Jorgensen-Thurston theory.

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned.

This book presents a comprehensive framework for the theoretical analysis of space-air-ground networks using stochastic geometry. This analytical approach is indispensable for evaluating the performance of large-scale space-air-ground networks, which serve as critical facilitators for the advancement of the sixth-generation wireless communication aimed at providing high-speed broadband coverage for remote areas. By incorporating the features of topology and channel model in different tiers, this book investigates the key performance metrics in terms of load balancing, coverage assessment, and mobility management. The developed mechanisms provide effective design insights for space-air-ground networks while obviating the need for complex system-level simulations.

Gravitation, electromagnetics and the two types of nuclear forces constitute the four fundamental forces of nature which regulate our everyday life. Amazingly, they are all described by a single idea of the 19th century proposed by Bernhard Riemann, and with the exception of gravitation, these ideas have been since confirmed by high energy experiments and cosmological observations. Geometry of the Fundamental Interactions - On Riemann's Legacy to High Energy Physics and Cosmology is a mathematical narrative of how we have come to agree on such a complex plot of nature, starting with the basic geometrical concepts and ending with hints on the perspective for cosmology.