These notes arise from lectures presented in Florence under the auspices of the Accadamia dei Lincee and deal with an area that lies at the crossroads of mathematics and physics. The material presented here rests primarily on the pioneering work of Vaughan Jones and Edward Witten relating polynomial invariants of knots to a topological quantum field theory in 2+1 dimensions. Professor Atiyah here presents an introduction to Witten's ideas from the mathematical point of view. The book will be essential reading for all geometers and gauge theorists as an exposition of new and interesting ideas in a rapidly developing area.

"The exposition is admirably lucid and elegant and I cannot imagine a better introduction, for graduate students and the more experienced, for mathematicians and physicists, to the large amount of high-powered mathematics required...As a source of insight this book is marvellous." Nature

"For those with some of the background...Atiyah's book is a marvelous appetizer, an invitation to delve more deeply. For the interested scientist who is not directly involved, there is still the pleasure of the safari in the hands of an excellent guide." American Scientist

"...provides an invaluable commentary for mathematicians to Witten's important paper 'Quantum field theory and the Jones polynomial'...." Steven Rosenberg, Mathematical Reviews

"The exposition is taut and the writing simple, elegant, and to the point. It is in fact a tour de force of single-minded and jargon-free exposition....we are dealing here more with poetry and inspirational writing than with the prose of everyday mathematics, and in this spirit, it is a pleasure to recommend this little volume to one and all." Raoul Bott, Bulletin of the American Mathematical Society