Many of the modern variational problems in topology arise in different but overlapping fields of scientific study: mechanics, physics and mathematics. In this work, Professor Fomenko offers a concise and clean explanation of some of these problems (both solved and unsolved), using current methods and analytical topology. The author's skillful exposition gives an unusual motivation to the theory expounded, and his work is recommended reading for specialists and nonspecialists alike, involved in the fields of physics and mathematics at both undergraduate and graduate levels.

Professor Anatolii Fomenko was educated at Moscow State University. He earned his DSc in 1972, and in 1974 he won the Moscow Mathematical Society Award for his doctoral thesis. Professor Fomenko has obtained fundamental results in the fields of geometry, topology and multidimensional variational calculus, and is also a successful teacher and specialist in scientific methodology.

Many of the modern variational problems of topology arise in different but overlapping fields of scientific study: mechanics, physics and mathematics. In this work, Professor Fomenko offers a concise and clear explanation of some of these problems (both solved and unsolved), using current methods of analytical topology. His book falls into three interrelated sections. The first gives an elementary introduction to some of the most important concepts of topology used in modern physics and mechanics: homology and cohomology, and fibration. The second investigates the significant role of Morse theory in modern aspects of the topology of smooth manifolds, particularly those of three and four dimensions. The third discusses minimal surfaces and harmonic mappings, and presents a number of classic physical experiments that lie at the foundations of modern understanding of multidimensional variational calculus. The author's skilful exposition of these topics and his own graphic illustrations give an unusual motivation to the theory expounded, and his work is recommended reading for specialists and non-specialists alike, involved in the fields of physics and mathematics at both undergraduate and graduate levels.

Preface, Chapter I. PRELIMINARIES, Chapter II. FUNCTIONS ON MANIFOLDS, Chapter III. MANIFOLDS OF SMALL DIMENSIONS, Chapter IV. MINIMAL SURFACES, References, Index