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Introducing foundational concepts in infinite-dimensional differential geometry beyond Banach manifolds, this text is based on Bastiani calculus. It focuses on two main areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry, exploring their connections to manifolds of (smooth) mappings. Topics covered include diffeomorphism groups, loop groups and Riemannian metrics for shape analysis. Numerous examples highlight both surprising connections between finite- and infinite-dimensional geometry, and challenges occurring solely in infinite dimensions. The geometric techniques developed are then showcased in modern applications of geometry such as geometric hydrodynamics, higher geometry in the guise of Lie groupoids, and rough path theory. With plentiful exercises, some with solutions, and worked examples, this will be indispensable for graduate students and researchers working at the intersection of functional analysis, non-linear differential equations and differential geometry. This title is also available as Open Access on Cambridge Core.

Infinite-dimensional manifolds. --- Geometry, Differential. --- Differential geometry --- Manifolds, Infinite-dimensional --- Global analysis (Mathematics) --- Topological manifolds

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This is the first existing volume that collects lectures on this important and fast developing subject in mathematics. The lectures are given by leading experts in the field and the range of topics is kept as broad as possible by including both the algebraic and the differential aspects of noncommutative geometry as well as recent applications to theoretical physics and number theory. Sample Chapter(s)
A Walk in the Noncommutative Garden (1,639 KB)

Contents:

• A Walk in the Noncommutative Garden (A Connes & M Marcolli)
• Renormalization of Noncommut

Noncommutative differential geometry. --- Differential geometry, Noncommutative --- Geometry, Noncommutative differential --- Non-commutative differential geometry --- Infinite-dimensional manifolds --- Operator algebras --- Differential equations

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In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive p-adic groups, Shimura varieties, the local Lfactors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect.

Noncommutative differential geometry --- Number theory --- Differential geometry, Noncommutative --- Geometry, Noncommutative differential --- Non-commutative differential geometry --- Infinite-dimensional manifolds --- Operator algebras --- Geometry. --- Mathematics --- Euclid's Elements

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This is an introduction to non-commutative geometry, with special emphasis on those cases where the structure algebra, which defines the geometry, is an algebra of matrices over the complex numbers. Applications to elementary particle physics are also discussed. This second edition is thoroughly revised and includes new material on reality conditions and linear connections plus examples from Jordanian deformations and quantum Euclidean spaces. Only some familiarity with ordinary differential geometry and the theory of fibre bundles is assumed, making this book accessible to graduate students and newcomers to this field.

Geometry, Algebraic. --- Noncommutative differential geometry. --- Differential geometry, Noncommutative --- Geometry, Noncommutative differential --- Non-commutative differential geometry --- Infinite-dimensional manifolds --- Operator algebras --- Algebraic geometry --- Geometry --- Noncommutative differential geometry --- Geometry, Algebraic

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Infinite-dimensional manifolds. --- Global analysis (Mathematics) --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Manifolds, Infinite-dimensional --- Topological manifolds