主编推荐 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第 一部教科书。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。
精彩内容
本书作者是Patrick Iglesi as-Zemnour是法国马赛数学研究所研究员(2019年退休),目前是以色列耶路撒冷希伯来大学常期的客座教授。主要从事辛几何和厂义流形的研究。2013年在美国数学会Mathematical Surveys and Monographs系列丛书第一次发表了关于广义流形的系统研究的专著。《广义流形》是该学科的第一本教科书,由美国数学协会出版,奠定了在理论物理中使用的微分几何主要领域的基础:可微性、卡坦微分学、同源和上同源、不同群、纤维束和连接等。书中还配有习题和解答有助于读者更好地学习。
媒体评论 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 that are or 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 operations: sum, product, subset and quotient. 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.
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