Visual Geometry and Topology

The principal aim of this book is to narrate, in anaccessible and fairly visual language, about some classicaland modern achievements of geometry and topology in bothintrinsic mathematical problems and applications tomathematical physics.

Geometry and topology are strongly motivated by thevisualization of ideal objects that have certain specialcharacteristics. A clear formulation of a specific propertyor a logically consistent proof of a theorem often comesonly after the mathematician has correctly "seen" what isgoing on. These pictures which are meant to serve assignposts leading to mathematical understanding, frequentlyalso contain a beauty of their own. The principal aim of this book is to narrate, in anaccessible and fairly visual language, about some classicaland modern achievements of geometry and topology in bothintrinsic mathematical problems and applications tomathematical physics. The book starts from classical notionsof topology and ends with remarkable new results inHamiltonian geometry. Fomenko lays special emphasis uponvisual explanations of the problems and results anddownplays the abstract logical aspects of calculations. Asan example, readers can very quickly penetrate into the newtheory of topological descriptions of integrable Hamiltoniandifferential equations. The book includes numerous graphicalsheets drawn by the author, which are presented in specialsections of "Visual material". These pictures illustrate themathematical ideas and results contained in the book. Usingthese pictures, the reader can understand many modernmathematical ideas and methods. Although "Visual Geometry and Topology" is aboutmathematics, Fomenko has written and illustrated this bookso that students and researchers from all the naturalsciences and also artists and art students will findsomething of interest within its pages.