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音像广义微分几何PatrickIglesias-Zemmour
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Preface
1. Diffeology and Diffeological Spaces
2. Locality and Diffeologies
3. Diffeological Vector Spaces
4. Modeling Spaces, Manifolds, etc.
5. Homotopy of Diffeological Spaces
6. Cartan-De Rham Calculus
7. Diffeological Groups
8. Diffeological Fiber Bundles
9. Symplectic Diffeology
Solutions to Exercises
Afterword
Notation and Vocabulary
Index
Bibliography
帕特里克·伊格莱西亚斯-泽穆尔(PatrickIglesias-Zemmour)是法国马赛数学研究所研究员,也是以色列希伯来大学的长期客座教授。他以辛几何和广义微分几何的研究而闻名。他所著的《广义微分几何》(Diffeology)是该领域全世界的部教材。
Diffeology is a natural extension of differential geometry that covers a wide spectrum of objects, ranging from singular spaces of any kind to infinite dimensional spaces, and sometimes mixing the two. With its developments in higher homotopy theory, fiber bundles, modeling spaces, Cartan-de-Rham calculus, moment map and symplectic constructions, to begin with, diffeology spans a wide range of traditional fields, treating geometrical objects tharer are not manifolds on an equal footing in a common framework.
This textbook establishes the foundations of the theory, and develops the main areas of application. These are illustrated by a series of examples, chosen explicitly because they are not covered by traditional differential geometry.
The advantage of diffeology comes from the conjunction of two strong properties: first, diffeology is stable by all set-theoretic oraios: sum, product, subsendqutient. It is said to be a complete and co-complete category. It is also Cartesian closed, the set of smooth maps having itself a natural diffeology. Second, and perhaps more importantly, diffeology even deals with non-Hausdorff quotient spaces in a non-trivial way, as is the case with irrational tori. This specific property becomes crucial for new constructions and is at the origin of the generalization of theorems that do not exist otherwise.
本书是国际上Diffeology研究领军人物Patrick Iglesias-Zemmour所撰写的经典名著,是这一领域世界上的 部教科书。原书于2013年由美国数学学会(AMS)作为Mathematical Surveys and Monographs系列丛书中的一卷出版。此次世界图书出版公司2022年所出的版本由作者亲自对全书做了全面修订,并专门写了修订版序,指出了修改的地方及首版出版后该领域的一些新进展。作者另外还有一本与本书配套的新作《广义微分几何讲义》(Lectures on Diffeology)也将由世界图书出版公司出版。
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