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Diffeology
Patrick Iglesias-Zemmour, CNRS, Marseille, France, and The Hebrew University of Jerusalem, Israel
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Mathematical Surveys and Monographs
2013; 439 pp; hardcover
Volume: 185
ISBN-10: 0-8218-9131-6
ISBN-13: 978-0-8218-9131-5
List Price: US$117 Member Price: US$93.60
Order Code: SURV/185

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Diffeology is an extension of differential geometry. With a minimal set of axioms, diffeology allows us to deal simply but rigorously with objects which do not fall within the usual field of differential geometry: quotients of manifolds (even non-Hausdorff), spaces of functions, groups of diffeomorphisms, etc. The category of diffeology objects is stable under standard set-theoretic operations, such as quotients, products, coproducts, subsets, limits, and colimits. With its right balance between rigor and simplicity, diffeology can be a good framework for many problems that appear in various areas of physics.

Actually, the book lays the foundations of the main fields of differential geometry used in theoretical physics: differentiability, Cartan differential calculus, homology and cohomology, diffeological groups, fiber bundles, and connections. The book ends with an open program on symplectic diffeology, a rich field of application of the theory. Many exercises with solutions make this book appropriate for learning the subject.

Graduate students and research mathematicians interested in differential geometry and mathematical physics.