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The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.

differentiaalvergelijkingen --- Partial differential equations --- Differential geometry. Global analysis --- differentiaal geometrie --- Symplectic geometry. --- Géométrie symplectique --- Global differential geometry. --- Differential equations, partial. --- Differential Geometry. --- Partial Differential Equations. --- Geometry, Differential --- Differential geometry. --- Partial differential equations. --- Differential geometry

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Since the early part of the 20th century, topology has gradually spread to many other branches of mathematics, and this book demonstrates how the subject continues to play a central role in the field. Written by a world-renowned mathematician, this classic text traces the history of algebraic topology beginning with its creation in the early 1900s and describes in detail the important theories that were discovered before 1960. Through the work of Poincaré, de Rham, Cartan, Hureqicz, and many others, this historical book also focuses on the emergence of new ideas and methods that have led 21st-century mathematicians towards new research directions. ***************************** This book is a well-informed and detailed analysis of the problems and development of algebraic topology, from Poincaré and Brouwer to Serre, Adams, and Thom. The author has examined each significant paper along this route and describes the steps and strategy of its proofs and its relation to other work. Previously, the history of the many technical developments of 20th-century mathematics had seemed to present insuperable obstacles to scholarship. This book demonstrates in the case of topology how these obstacles can be overcome, with enlightening results.... Within its chosen boundaries the coverage of this book is superb. Read it! —MathSciNet [The author] traces the development of algebraic and differential topology from the innovative work by Poincaré at the turn of the century to the period around 1960. [He] has given a superb account of the growth of these fields.… The details are interwoven with the narrative in a very pleasant fashion.… [The author] has previous written histories of functional analysis and of algebraic geometry, but neither book was on such a grand scale as this one. He has made it possible to trace the important steps in the growth of algebraic and differential topology, and to admire the hard work and major advances made by the founders. —Zentralblatt MATH.

Algebraic topology -- History. --- Differential topology -- History. --- Algebraic topology --- Differential topology --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- History --- 515.14 --- -Differential topology --- -Geometry, Differential --- -Algebraic topology --- 515.14 Algebraic topology --- -515.14 Algebraic topology --- Mathematics. --- Differential geometry. --- History. --- Algebraic topology. --- Algebraic Topology. --- Differential Geometry. --- History of Mathematical Sciences. --- Geometry, Differential --- Topology --- Annals --- Auxiliary sciences of history --- Differential geometry --- Math --- Science --- anno 1900-1999 --- Global differential geometry. --- Algebraic topology - History --- Differential topology - History

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Arising from the fourth Dagstuhl conference entitled Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data (2011), this book offers a broad and vivid view of current work in this emerging field. Topics covered range from applications of the analysis of tensor fields to research on their mathematical and analytical properties. Part I, Tensor Data Visualization, surveys techniques for visualization of tensors and tensor fields in engineering, discusses the current state of the art and challenges, and examines tensor invariants and glyph design, including an overview of common glyphs. The second Part, Representation and Processing of Higher-order Descriptors, describes a matrix representation of local phase, outlines mathematical morphological operations techniques, extended for use in vector images, and generalizes erosion to the space of diffusion weighted MRI. Part III, Higher Order Tensors and Riemannian-Finsler Geometry, offers powerful mathematical language to model and analyze large and complex diffusion data such as High Angular Resolution Diffusion Imaging (HARDI) and Diffusion Kurtosis Imaging (DKI). A Part entitled Tensor Signal Processing presents new methods for processing tensor-valued data, including a novel perspective on performing voxel-wise morphometry of diffusion tensor data using kernel-based approach, explores the free-water diffusion model, and reviews proposed approaches for computing fabric tensors, emphasizing trabecular bone research. The last Part, Applications of Tensor Processing, discusses metric and curvature tensors, two of the most studied tensors in geometry processing. Also covered is a technique for diagnostic prediction of first-episode schizophrenia patients based on brain diffusion MRI data. The last chapter presents an interactive system integrating the visual analysis of diffusion MRI tractography with data from electroencephalography.

Calculus of tensors. --- Calculus of tensors --- Information visualization. --- Data processing. --- Data visualization --- Visualization of information --- Information science --- Visual analytics --- Absolute differential calculus --- Analysis, Tensor --- Calculus, Absolute differential --- Calculus, Tensor --- Tensor analysis --- Tensor calculus --- Geometry, Differential --- Geometry, Infinitesimal --- Vector analysis --- Spinor analysis --- Visualization. --- Differential equations, partial. --- Global differential geometry. --- Computer vision. --- Computer graphics. --- Partial Differential Equations. --- Differential Geometry. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Computer Graphics. --- Theoretical, Mathematical and Computational Physics. --- Automatic drafting --- Graphic data processing --- Graphics, Computer --- Computer art --- Graphic arts --- Electronic data processing --- Engineering graphics --- Image processing --- Machine vision --- Vision, Computer --- Artificial intelligence --- Pattern recognition systems --- Partial differential equations --- Visualisation --- Imagination --- Visual perception --- Imagery (Psychology) --- Digital techniques --- Mathematics. --- Partial differential equations. --- Differential geometry. --- Optical data processing. --- Mathematical physics. --- Optical computing --- Visual data processing --- Bionics --- Integrated optics --- Photonics --- Computers --- Differential geometry --- Physical mathematics --- Physics --- Math --- Science --- Optical equipment --- Mathematics