Geometry and topology are strongly motivated by the visualization of ideal objects that have certain special characteristics. A clear formulation of a specific property or a logically consistent proof of a theorem often comes only after the mathematician has correctly "seen" what is going on. These pictures which are meant to serve as signposts leading to mathematical understanding, frequently also contain a beauty of their own. The principal aim of this book is to narrate, in an accessible and fairly visual language, about some classical and modern achievements of geometry and topology in both intrinsic mathematical problems and applications to mathematical physics. The book starts from classical notions of topology and ends with remarkable new results in Hamiltonian geometry. Fomenko lays special emphasis upon visual explanations of the problems and results and downplays the abstract logical aspects of calculations. As an example, readers can very quickly penetrate into the new theory of topological descriptions of integrable Hamiltonian differential equations. The book includes numerous graphical sheets drawn by the author, which are presented in special sections of "Visual material". These pictures illustrate the mathematical ideas and results contained in the book. Using these pictures, the reader can understand many modern mathematical ideas and methods. Although "Visual Geometry and Topology" is about mathematics, Fomenko has written and illustrated this book so that students and researchers from all the natural sciences and also artists and art students will find something of interest within its pages.
Author(s): Anatolij T. Fomenko
Edition: 1
Publisher: Springer
Year: 2011
Language: English
Pages: 340
City: Berlin
Tags: Geometry; Hamiltonian Systems; Hamiltonsche Systeme; Integrable Systems; Integrierbare Systeme; Visual Geometry; Visual Topology; Visuelle Geometrie; Visuelle Topologie; Mathematical Physics; Topology
Front Matter
Pages I-XVI
Polyhedra. Simplicial Complexes. Homologies
Anatolij T. Fomenko
Pages 1-73
Low-Dimensional Manifolds
Anatolij T. Fomenko
Pages 75-191
Visual Symplectic Topology and Visual Hamiltonian Mechanics
Anatolij T. Fomenko
Pages 193-254
Visual Images in Some Other Fields of Geometry and in Its Applications
Anatolij T. Fomenko
Pages 255-292
Back Matter
Pages 293-324
