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This compression is particularly true of the new section on ' Gauge theoretical physics' added for the English edition. Abstract ideas in geometry are now intensively studied as models for the physics of elementary particles. Index theory is central here, particularly the analytical 'heat equation' proof, which is briefly mentioned in the book. This last section contains quantum mechanics and field theory, Riemannian geometry and connections, all packed in together. The material is not so well polished as the earlier sections and much of it is available in standard references, but I found the introductory paragraphs 'Physical motivation and overview' very stimulating.

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