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This book aims to provide a friendly introduction to non-commutative geometry. It studies index theory from a classical differential geometry perspective up to the point where classical differential geometry methods become insufficient. It then presents non-commutative geometry as a natural continuation of classical differential geometry. It thereby aims to provide a natural link between classical differential geometry and non-commutative geometry. The book shows that the index formula is a topological statement, and ends with non-commutative topology.

Differential geometry. --- Manifolds (Mathematics). --- Complex manifolds. --- Differential Geometry. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Differential geometry --- Geometry, Differential.

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Edited in collaboration with the Grassmann Research Group, this book contains many important articles delivered at the ICM 2014 Satellite Conference and the 18th International Workshop on Real and Complex Submanifolds, which was held at the National Institute for Mathematical Sciences, Daejeon, Republic of Korea, August 10–12, 2014. The book covers various aspects of differential geometry focused on submanifolds, symmetric spaces, Riemannian and Lorentzian manifolds, and Kähler and Grassmann manifolds.

Submanifolds --- Geometry, Differential --- Manifolds (Mathematics) --- Global differential geometry. --- Cell aggregation --- Differential Geometry. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Mathematics. --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Differential geometry. --- Manifolds (Mathematics). --- Complex manifolds. --- Analytic spaces --- Topology --- Differential geometry

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Among all the Hamiltonian systems, the integrable ones have special geometric properties; in particular, their solutions are very regular and quasi-periodic. The quasi-periodicity of the solutions of an integrable system is a result of the fact that the system is invariant under a (semi-global) torus action. It is thus natural to investigate the symplectic manifolds that can be endowed with a (global) torus action. This leads to symplectic toric manifolds (Part B of this book). Physics makes a surprising come-back in Part A: to describe Mirror Symmetry, one looks for a special kind of Lagrangian submanifolds and integrable systems, the special Lagrangians. Furthermore, integrable Hamiltonian systems on punctured cotangent bundles are a starting point for the study of contact toric manifolds (Part C of this book).

Differential geometry. Global analysis --- Differential geometry. --- Manifolds (Mathematics). --- Complex manifolds. --- Physics. --- Differential Geometry. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Mathematical Methods in Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Differential geometry

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Written by leading experts in the field, this monograph provides homotopy theoretic foundations for surgery theory on higher-dimensional manifolds. Presenting classical ideas in a modern framework, the authors carefully highlight how their results relate to (and generalize) existing results in the literature. The central result of the book expresses algebraic surgery theory in terms of the geometric Hopf invariant, a construction in stable homotopy theory which captures the double points of immersions. Many illustrative examples and applications of the abstract results are included in the book, making it of wide interest to topologists. Serving as a valuable reference, this work is aimed at graduate students and researchers interested in understanding how the algebraic and geometric topology fit together in the surgery theory of manifolds. It is the only book providing such a wide-ranging historical approach to the Hopf invariant, double points and surgery theory, with many results old and new. .

Homotopy theory. --- Algebraic topology. --- Manifolds (Mathematics) --- Invariant manifolds. --- Deformations, Continuous --- Mathematics. --- Manifolds (Mathematics). --- Complex manifolds. --- Algebraic Topology. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Topology --- Geometry, Differential --- Invariants --- Cell aggregation --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Analytic spaces

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"This book contains essential material that every graduate student must know. Written with Serge Lang's inimitable wit and clarity, the volume introduces the reader to manifolds, differential forms, Darboux's theorem, Frobenius, and all the central features of the foundations of differential geometry. Lang lays the basis for further study in geometric analysis, and provides a solid resource in the techniques of differential topology. The book will have a key position on my shelf. Steven Krantz, Washington University in St. Louis "This is an elementary, finite dimensional version of the author's classic monograph, Introduction to Differentiable Manifolds (1962), which served as the standard reference for infinite dimensional manifolds. It provides a firm foundation for a beginner's entry into geometry, topology, and global analysis. The exposition is unencumbered by unnecessary formalism, notational or otherwise, which is a pitfall few writers of introductory texts of the subject manage to avoid. The author's hallmark characteristics of directness, conciseness, and structural clarity are everywhere in evidence. A nice touch is the inclusion of more advanced topics at the end of the book, including the computation of the top cohomology group of a manifold, a generalized divergence theorem of Gauss, and an elementary residue theorem of several complex variables. If getting to the main point of an argument or having the key ideas of a subject laid bare is important to you, then you would find the reading of this book a satisfying experience." Hung-Hsi Wu, University of California, Berkeley.

Differential topology. --- Differentiable manifolds. --- Topology. --- Cell aggregation --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Mathematics. --- Manifolds (Mathematics). --- Complex manifolds. --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear