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Differential Geometry


Characteristic Classes. (AM-76)

Characteristic Classes. (AM-76) Lowest new price: $78.52
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Author: John Milnor
Brand: Princeton University Press

The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds.



In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers.



Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.

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A Course in Minimal Surfaces (Graduate Studies in Mathematics)

A Course in Minimal Surfaces (Graduate Studies in Mathematics) Lowest new price: $67.00
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Author: Tobias Holck Colding

Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), the authors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science. The only prerequisites needed for this book are a basic knowledge of Riemannian geometry and some familiarity with the maximum principle.

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Some Nonlinear Problems in Riemannian Geometry (Springer Monographs in Mathematics)

Some Nonlinear Problems in Riemannian Geometry (Springer Monographs in Mathematics) Lowest new price: $128.00
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Author: Thierry Aubin

This book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, geometric and topological methods. It explores important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved.

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Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields (Theoretical and Mathematical Physics)

Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields (Theoretical and Mathematical Physics) Lowest new price: $153.07
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Author: Gerd Rudolph

The book is devoted to the study of the geometrical and topological structure of gauge theories. It consists of the following three building blocks:
- Geometry and topology of fibre bundles,
- Clifford algebras, spin structures and Dirac operators,
- Gauge theory.
Written in the style of a mathematical textbook, it combines a comprehensive presentation of the mathematical foundations with a discussion of a variety of advanced topics in gauge theory.
The first building block includes a number of specific topics, like invariant connections, universal connections, H-structures and the Postnikov approximation of classifying spaces.
Given the great importance of Dirac operators in gauge theory, a complete proof of the Atiyah-Singer Index Theorem is presented. The gauge theory part contains the study of Yang-Mills equations (including the theory of instantons and the classical stability analysis), the discussion of various models with matter fields (including magnetic monopoles, the Seiberg-Witten model and dimensional reduction) and the investigation of the structure of the gauge orbit space. The final chapter is devoted to elements of quantum gauge theory including the discussion of the Gribov problem, anomalies and the implementation of the non-generic gauge orbit strata in the framework of Hamiltonian lattice gauge theory.
The book is addressed both to physicists and mathematicians. It is intended to be accessible to students starting from a graduate level.

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Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics)

Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics) Lowest new price: $83.00
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Author: Sigurdur Helgason
Brand: Brand: American Mathematical Society

The study of homogeneous spaces provides excellent insights into both differential geometry and Lie groups. In geometry, for instance, general theorems and properties will also hold for homogeneous spaces, and will usually be easier to understand and to prove in this setting. For Lie groups, a significant amount of analysis either begins with or reduces to analysis on homogeneous spaces, frequently on symmetric spaces. For many years and for many mathematicians, Sigurdur Helgason's classic Differential Geometry, Lie Groups, and Symmetric Spaces has been--and continues to be--the standard source for this material.

Helgason begins with a concise, self-contained introduction to differential geometry. He then introduces Lie groups and Lie algebras, including important results on their structure. This sets the stage for the introduction and study of symmetric spaces, which form the central part of the book. The text concludes with the classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over $\mathbf{C}$ and Cartan's classification of simple Lie algebras over $\mathbf{R}$.

The excellent exposition is supplemented by extensive collections of useful exercises at the end of each chapter. All the problems have either solutions or substantial hints, found at the back of the book.

For this latest edition, Helgason has made corrections and added helpful notes and useful references. The sequels to the present book are published in the AMS's Mathematical Surveys and Monographs Series: Groups and Geometric Analysis, Volume 83, and Geometric Analysis on Symmetric Spaces, Volume 39.

Sigurdur Helgason was awarded the Steele Prize for Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis.

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Geometric Analysis of the Bergman Kernel and Metric (Graduate Texts in Mathematics)

Geometric Analysis of the Bergman Kernel and Metric (Graduate Texts in Mathematics) Lowest new price: $51.63
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Author: Steven G. Krantz

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.


A Visual Introduction to Differential Forms and Calculus on Manifolds

A Visual Introduction to Differential Forms and Calculus on Manifolds Lowest new price: $79.99
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Author: Jon Pierre Fortney

This book explains and helps readers to develop geometric intuition as it relates to differential forms. It includes over 250 figures to aid understanding and enable readers to visualize the concepts being discussed. The author gradually builds up to the basic ideas and concepts so that definitions, when made, do not appear out of nowhere, and both the importance and role that theorems play is evident as or before they are presented. With a clear writing style and easy-to- understand motivations for each topic, this book is primarily aimed at second- or third-year undergraduate math and physics students with a basic knowledge of vector calculus and linear algebra.

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A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition

A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition Lowest new price: $40.00
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Author: Michael Spivak
Brand: Brand: Publish or Perish

Spivak's 5 volume set is a classic and overall the best and most thorough treatment of differential geometry

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Gauge Theory of Elementary Particle Physics: Problems and Solutions

Gauge Theory of Elementary Particle Physics: Problems and Solutions Lowest new price: $76.80
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Author: Ta-Pei Cheng

Students of particle physics often find it difficult to locate resources to learn calculational techniques as intermediate steps are usually not given in the research literature. To a certain extent, this is the case even in some of the established textbooks. In this book of worked problems enough details are provided so that the beginner will understand the solution in each particular case. With this step-by-step guidance, students (after first attempting the solution themselves) can develop their skills and confidence in their ability to work out particle theory problems. Besides being a problems/solutions addition to the well established textbook by Cheng & Li, this book introduces several new topics. It provides the reader with a self-contained approach to the subject, suitable even for those not familiar with the textbook. All problems have been given a descriptive title, enabling the reader to select according to his preferences.

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A Comprehensive Introduction to Differential Geometry, Vol. 5, 3rd Edition

A Comprehensive Introduction to Differential Geometry, Vol. 5, 3rd Edition Lowest new price: $50.00
Lowest used price: $38.51
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Author: Michael Spivak
Brand: Brand: Publish or Perish

Spivak's 5 volume set is a classic and overall the best and most thorough treatment of differential geometry

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