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Topology --- Function spaces. --- Topology. --- Dimension theory (Topology) --- Infinite-dimensional manifolds.

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This book constitutes the proceedings of the 2000 Howard conference on "Infinite Dimensional Lie Groups in Geometry and Representation Theory". It presents some important recent developments in this area. It opens with a topological characterization of regular groups, treats among other topics the integrability problem of various infinite dimensional Lie algebras, presents substantial contributions to important subjects in modern geometry, and concludes with interesting applications to representation theory. The book should be a new source of inspiration for advanced graduate students and esta

Infinite dimensional Lie algebras --- Infinite groups --- Infinite-dimensional manifolds --- Lie groups

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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