A Course in Metric Geometry

Author: Dmitri Burago,I͡Uriĭ Dmitrievich Burago,Sergeĭ Ivanov

Publisher: American Mathematical Soc.

ISBN: 0821821296

Category: Mathematics

Page: 415

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``Metric geometry'' is an approach to geometry based on the notion of length on a topological space. This approach experienced a very fast development in the last few decades and penetrated into many other mathematical disciplines, such as group theory, dynamical systems, and partial differential equations. The objective of this graduate textbook is twofold: to give a detailed exposition of basic notions and techniques used in the theory of length spaces, and, more generally, to offer an elementary introduction into a broad variety of geometrical topics related to the notion of distance, including Riemannian and Carnot-Caratheodory metrics, the hyperbolic plane, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic spaces, convergence of metric spaces, and Alexandrov spaces (non-positively and non-negatively curved spaces). The authors tend to work with ``easy-to-touch'' mathematical objects using ``easy-to-visualize'' methods. The authors set a challenging goal of making the core parts of the book accessible to first-year graduate students. Most new concepts and methods are introduced and illustrated using simplest cases and avoiding technicalities. The book contains many exercises, which form a vital part of the exposition.

Einführung in die Kategorientheorie

Mit ausführlichen Erklärungen und zahlreichen Beispielen

Author: Martin Brandenburg

Publisher: Springer-Verlag

ISBN: 3662535211

Category: Mathematics

Page: 343

View: 8995

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Die Kategorientheorie deckt die innere Architektur der Mathematik auf. Dabei werden die strukturellen Gemeinsamkeiten zwischen mathematischen Disziplinen und ihren spezifischen Konstruktionen herausgearbeitet. Dieses Buch gibt eine systematische Einführung in die Grundbegriffe der Kategorientheorie. Zahlreiche ausführliche Erklärungstexte sowie die große Menge an Beispielen helfen beim Einstieg in diese verhältnismäßig abstrakte Theorie. Es werden viele konkrete Anwendungen besprochen, welche die Nützlichkeit der Kategorientheorie im mathematischen Alltag belegen. Jedes Kapitel wird mit einem motivierenden Text eingeleitet und mit einer großen Aufgabensammlung abgeschlossen. An Vorwissen muss der Leser lediglich ein paar Grundbegriffe des Mathematik-Studiums mitbringen. Die vorliegende zweite vollständig durchgesehene Auflage ist um ausführliche Lösungen zu ausgewählten Aufgaben ergänzt.

A Course in Differential Geometry

Author: Thierry Aubin

Publisher: American Mathematical Soc.

ISBN: 9780821872147

Category: Mathematics

Page: 184

View: 6679

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This textbook for second-year graduate students is an introduction to differential geometry with principal emphasis on Riemannian geometry. The author is well-known for his significant contributions to the field of geometry and PDEs - particularly for his work on the Yamabe problem - and for his expository accounts on the subject. The text contains many problems and solutions, permitting the reader to apply the theorems and to see concrete developments of the abstract theory.

Numerical Geometry of Non-Rigid Shapes

Author: Alexander M. Bronstein,Michael M. Bronstein,Ron Kimmel

Publisher: Springer Science & Business Media

ISBN: 9780387733012

Category: Computers

Page: 346

View: 7573

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Deformable objects are ubiquitous in the world surrounding us, on all levels from micro to macro. The need to study such shapes and model their behavior arises in a wide spectrum of applications, ranging from medicine to security. In recent years, non-rigid shapes have attracted growing interest, which has led to rapid development of the field, where state-of-the-art results from very different sciences - theoretical and numerical geometry, optimization, linear algebra, graph theory, machine learning and computer graphics, to mention several - are applied to find solutions. This book gives an overview of the current state of science in analysis and synthesis of non-rigid shapes. Everyday examples are used to explain concepts and to illustrate different techniques. The presentation unfolds systematically and numerous figures enrich the engaging exposition. Practice problems follow at the end of each chapter, with detailed solutions to selected problems in the appendix. A gallery of colored images enhances the text. This book will be of interest to graduate students, researchers and professionals in different fields of mathematics, computer science and engineering. It may be used for courses in computer vision, numerical geometry and geometric modeling and computer graphics or for self-study.

Interacting Stochastic Systems

Author: Jean-Dominique Deuschel,Andreas Greven

Publisher: Springer Science & Business Media

ISBN: 3540271104

Category: Mathematics

Page: 450

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Core papers emanating from the research network, DFG-Schwerpunkt: Interacting stochastic systems of high complexity.

Manifolds and Differential Geometry

Author: Jeffrey Marc Lee

Publisher: American Mathematical Soc.

ISBN: 0821848151

Category: Mathematics

Page: 671

View: 9943

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Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle. This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hyper-surfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations. The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry.

Introduction to Hilbert Space

Author: Sterling K. Berberian

Publisher: American Mathematical Soc.

ISBN: 0821819127

Category: Mathematics

Page: 206

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Completely self-contained ... All proofs are given in full detail ... recommended for unassisted reading by beginners ... For teaching purposes this book is ideal. --Proceedings of the Edinburgh Mathematical Society The book is easy to read and, although the author had in mind graduate students, most of it is obviously appropriate for an advanced undergraduate course. It is also a book which a reasonably good student might read on his own. --Mathematical Reviews This textbook evolved from a set of course notes for first- or second-year graduate students in mathematics and related fields such as physics. It presents, in a self-contained way, various aspects of geometry and analysis of Hilbert spaces, including the spectral theorem for compact operators. Over 400 exercises provide examples and counter-examples for definitions and theorems in the book, as well as generalization of some material in the text. Aside from being an exposition of basic material on Hilbert space, this book may also serve as an introduction to other areas of functional analysis. The only prerequisite for understanding the material is a standard foundation in advanced calculus. The main notions of linear algebra, such as vector spaces, bases, etc., are explained in the first chapter of the book.

Quasiconformal Mappings and Their Applications

Author: Saminathan Ponnusamy,T. Sugawa,Matti Vuorinen

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: 354

View: 7751

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"Quasiconformal Mappings and their Applications covers conformal invariance and conformally invariant metrics, hyperbolic-type metrics and hyperbolic geodesics, isometries of relative metrics, uniform spaces and Gromov hyperbolicity, quasiregular mappings and quasiconformal mappings in n-space, universal Teichmuller space and related topics, quasiminimizers and potential theory, and numerical conformal mapping and circle packings."--BOOK JACKET.

A short course in differential geometry and topology

Author: A. T. Fomenko,Aleksandr Sergeevich Mishchenko

Publisher: N.A

ISBN: 9781904868323

Category: Mathematics

Page: 273

View: 878

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"This volume is intended for graduate and research students in mathematics and physics. It covers general topology, nonlinear co-ordinate systems, theory of smooth manifolds, theory of curves and surfaces, transformation groups, tensor analysis and Riemannian geometry, theory of integration and homologies, fundamental groups and variational principles in Riemannian geometry. The text is presented in a form that is easily accessible to students and is supplemented by a large number of examples, problems, drawings and appendices."--Cambridge Scientific Publishers website, viewed 2 September 2009.

A Course in Modern Mathematical Physics

Groups, Hilbert Space and Differential Geometry

Author: Peter Szekeres

Publisher: Cambridge University Press

ISBN: 1139455834

Category: Science

Page: N.A

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This book, first published in 2004, provides an introduction to the major mathematical structures used in physics today. It covers the concepts and techniques needed for topics such as group theory, Lie algebras, topology, Hilbert space and differential geometry. Important theories of physics such as classical and quantum mechanics, thermodynamics, and special and general relativity are also developed in detail, and presented in the appropriate mathematical language. The book is suitable for advanced undergraduate and beginning graduate students in mathematical and theoretical physics, as well as applied mathematics. It includes numerous exercises and worked examples, to test the reader's understanding of the various concepts, as well as extending the themes covered in the main text. The only prerequisites are elementary calculus and linear algebra. No prior knowledge of group theory, abstract vector spaces or topology is required.

Geometry & Topology

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Geometry

Page: N.A

View: 8122

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Fully refereed international journal dealing with all aspects of geometry and topology and their applications.

Geometric Analysis of the Bergman Kernel and Metric

Author: Steven G. Krantz

Publisher: Springer Science & Business Media

ISBN: 146147924X

Category: Mathematics

Page: 292

View: 3777

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This text provides a masterful and systematic treatment of all the basic analytic and geometric aspects of Bergman's classic theory of the kernel and its invariance properties. These include calculation, invariance properties, boundary asymptotics, and asymptotic expansion of the Bergman kernel and metric. Moreover, it presents a unique compendium of results with applications to function theory, geometry, partial differential equations, and interpretations in the language of functional analysis, with emphasis on the several complex variables context. Several of these topics appear here for the first time in book form. Each chapter includes illustrative examples and a collection of exercises which will be of interest to both graduate students and experienced mathematicians. Graduate students who have taken courses in complex variables and have a basic background in real and functional analysis will find this textbook appealing. Applicable courses for either main or supplementary usage include those in complex variables, several complex variables, complex differential geometry, and partial differential equations. Researchers in complex analysis, harmonic analysis, PDEs, and complex differential geometry will also benefit from the thorough treatment of the many exciting aspects of Bergman's theory.

Strasbourg Master Class on Geometry

Author: Athanase Papadopoulos

Publisher: European Mathematical Society

ISBN: 9783037191057

Category: Mathematics

Page: 454

View: 4623

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The text is addressed to students and mathematicians who wish to learn the subject. It can also be used as a reference book and as a textbook for short courses on geometry.

Compact Riemann Surfaces

An Introduction to Contemporary Mathematics

Author: Jürgen Jost

Publisher: Springer Science & Business Media

ISBN: 9783540330677

Category: Mathematics

Page: 282

View: 1114

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This book is novel in its broad perspective on Riemann surfaces: the text systematically explores the connection with other fields of mathematics. The book can serve as an introduction to contemporary mathematics as a whole, as it develops background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry. The book is unique among textbooks on Riemann surfaces in its inclusion of an introduction to Teichmüller theory. For this new edition, the author has expanded and rewritten several sections to include additional material and to improve the presentation.