Étienne Ghys - Böcker

An illustration-packed dive into the geometry, engineering, and physics of soccer ballsThe Football takes readers on an entertaining and fact-filled exploration of the mathematical secrets of the most popular spherical object on the planet. The football is familiar to billions of fans across the globe, but how many really look at it? Do footballs all have the same shape? Spoiler: not exactly. How does their shape affect how they play? With Étienne Ghys as our guide, we discover why ballistics, friction, and air flow are key to scoring goals—and why the football is a mathematical problem that engineers are still trying to solve.Ghys begins with the classic Telstar ball used in the 1970 World Cup in Mexico. Its twelve black pentagons and twenty white hexagons are what most of us picture when we think of the sport. Following the story through successive World Cups, he shows how engineers constantly challenge themselves to reinvent the ball, aiming for a perfect sphere while accounting for manufacturing requirements and aerodynamics. Along the way, Ghys introduces us to the mathematics of Platonic solids, symmetries, polyhedra, turbulence, roughness, drag, and spin. He paints engaging portraits of the engineers and sports insiders who study and apply these phenomena and explains how the skills of players factor into how the ball behaves, whether the game is being played in stadiums, schoolyards, or backyards.Featuring a wealth of color illustrations, The Football blends a lively narrative with insights from a world-renowned geometer to tell a mathematical story unlike any other.

The theory of hyperbolic groups has its starting point in a fundamental paper by M. Gromov, published in 1987. These are finitely generated groups that share important properties with negatively curved Riemannian manifolds. This monograph is intended to be an introduction to part of Gromov's theory, giving basic definitions, some of the most important examples, various properties of hyperbolic groups, and an application to the construction of infinite torsion groups. The main theme is the relevance of geometric ideas to the understanding of finitely generated groups. In addition to chapters written by the editors, contributions by W. Ballmann, A. Haefliger, E. Salem, R. Strebel, and M. Troyanov are also included.The book will be particularly useful to researchers in combinatorial group theory, Riemannian geometry, and theoretical physics, as well as post-graduate students interested in these fields.

Henri Poincare (1854-1912) was one of the greatest scientists of his time, perhaps the last one to have mastered and expanded almost all areas in mathematics and theoretical physics. He created new mathematical branches, such as algebraic topology, dynamical systems, and automorphic functions, and he opened the way to complex analysis with several variables and to the modern approach to asymptotic expansions. He revolutionized celestial mechanics, discovering deterministic chaos. In physics, he is one of the fathers of special relativity, and his work in the philosophy of sciences is illuminating. For this book, about twenty world experts were asked to present one part of Poincare's extraordinary work. Each chapter treats one theme, presenting Poincare's approach, and achievements, along with examples of recent applications and some current prospects. Their contributions emphasize the power and modernity of the work of Poincare, an inexhaustible source of inspiration for researchers, as illustrated by the Fields Medal awarded in 2006 to Grigori perelman for his proof of the Poincare conjecture stated a century before. This book can be read by anyone with a master's (even a bachelor's) degree in mathematics, or physics, or more generally by anyone who likes mathematical and physical ideas. Rather than presenting detailed proofs, the main ideas are explained, and a bibliography is provided for those who wish to understand the technical details.

This proceedings presents the latest research materials done on group theory from geometrical viewpoint in particular Gromov's theory of hyperbolic groups, Coxeter groups, Tits buildings and actions on real trees. All these are very active subjects.