# Search Results for "applications-of-contact-geometry-and-topology-in-physics"

## Applications of Contact Geometry and Topology in Physics

**Author**: Arkady Leonidovich Kholodenko**Publisher:**World Scientific**ISBN:**9814412090**Category:**Mathematics**Page:**492**View:**3731

Although contact geometry and topology is briefly discussed in V I Arnol''d''s book Mathematical Methods of Classical Mechanics (Springer-Verlag, 1989, 2nd edition), it still remains a domain of research in pure mathematics, e.g. see the recent monograph by H Geiges An Introduction to Contact Topology (Cambridge U Press, 2008). Some attempts to use contact geometry in physics were made in the monograph Contact Geometry and Nonlinear Differential Equations (Cambridge U Press, 2007). Unfortunately, even the excellent style of this monograph is not sufficient to attract the attention of the physics community to this type of problems. This book is the first serious attempt to change the existing status quo. In it we demonstrate that, in fact, all branches of theoretical physics can be rewritten in the language of contact geometry and topology: from mechanics, thermodynamics and electrodynamics to optics, gauge fields and gravity; from physics of liquid crystals to quantum mechanics and quantum computers, etc. The book is written in the style of famous Landau-Lifshitz (L-L) multivolume course in theoretical physics. This means that its readers are expected to have solid background in theoretical physics (at least at the level of the L-L course). No prior knowledge of specialized mathematics is required. All needed new mathematics is given in the context of discussed physical problems. As in the L-L course some problems/exercises are formulated along the way and, again as in the L-L course, these are always supplemented by either solutions or by hints (with exact references). Unlike the L-L course, though, some definitions, theorems, and remarks are also presented. This is done with the purpose of stimulating the interest of our readers in deeper study of subject matters discussed in the text.

## Physics, Geometry and Topology

**Author**: H.C. Lee**Publisher:**Springer Science & Business Media**ISBN:**1461538025**Category:**Science**Page:**681**View:**5495

The Banff NATO Summer School was held August 14-25, 1989 at the Banff Cen tre, Banff, Albert, Canada. It was a combination of two venues: a summer school in the annual series of Summer School in Theoretical Physics spon sored by the Theoretical Physics Division, Canadian Association of Physi cists, and a NATO Advanced Study Institute. The Organizing Committee for the present school was composed of G. Kunstatter (University of Winnipeg), H.C. Lee (Chalk River Laboratories and University of Western Ontario), R. Kobes (University of Winnipeg), D.l. Toms (University of Newcastle Upon Tyne) and Y.S. Wu (University of Utah). Thanks to the group of lecturers (see Contents) and the timeliness of the courses given, the school, entitled PHYSICS, GEOMETRY AND TOPOLOGY, was popular from the very outset. The number of applications outstripped the 90 places of accommodation reserved at the Banff Centre soon after the school was announced. As the eventual total number of participants was increased to 170, it was still necessary to tum away many deserving applicants. In accordance with the spirit of the school, the geometrical and topologi cal properties in each of the wide ranging topics covered by the lectures were emphasized. A recurring theme in a number of the lectures is the Yang-Baxter relation which characterizes a very large class of integrable systems including: many state models, two-dimensional conformal field theory, quantum field theory and quantum gravity in 2 + I dimensions.

## Geometry, Topology and Physics, Second Edition

**Author**: Mikio Nakahara**Publisher:**CRC Press**ISBN:**9780750306065**Category:**Mathematics**Page:**596**View:**8291

Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields. The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view. Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.

## Topology and Geometry for Physicists

**Author**: Charles Nash,Siddhartha Sen**Publisher:**Courier Corporation**ISBN:**0486318362**Category:**Mathematics**Page:**320**View:**8663

Differential geometry and topology are essential tools for many theoretical physicists, particularly in the study of condensed matter physics, gravity, and particle physics. Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations, enabling students to apply the techniques to their physics formulas and research. "Thoroughly recommended" by The Physics Bulletin, this volume's physics applications range from condensed matter physics and statistical mechanics to elementary particle theory. Its main mathematical topics include differential forms, homotopy, homology, cohomology, fiber bundles, connection and covariant derivatives, and Morse theory.

## An Introduction to Contact Topology

**Author**: Hansjörg Geiges**Publisher:**Cambridge University Press**ISBN:**1139467956**Category:**Mathematics**Page:**N.A**View:**6873

This text on contact topology is a comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.

## Topology and Geometry for Physics

**Author**: Helmut Eschrig**Publisher:**Springer**ISBN:**3642147003**Category:**Science**Page:**390**View:**1593

A concise but self-contained introduction of the central concepts of modern topology and differential geometry on a mathematical level is given specifically with applications in physics in mind. All basic concepts are systematically provided including sketches of the proofs of most statements. Smooth finite-dimensional manifolds, tensor and exterior calculus operating on them, homotopy, (co)homology theory including Morse theory of critical points, as well as the theory of fiber bundles and Riemannian geometry, are treated. Examples from physics comprise topological charges, the topology of periodic boundary conditions for solids, gauge fields, geometric phases in quantum physics and gravitation.

## Contact Geometry and Nonlinear Differential Equations

**Author**: Alexei Kushner,Valentin Lychagin,Vladimir Rubtsov**Publisher:**Cambridge University Press**ISBN:**0521824761**Category:**Mathematics**Page:**496**View:**8474

Shows novel and modern ways of solving differential equations using methods from contact and symplectic geometry.

## Basic Elements of Differential Geometry and Topology

**Author**: S.P. Novikov,A.T. Fomenko**Publisher:**Springer Science & Business Media**ISBN:**9401578958**Category:**Mathematics**Page:**490**View:**8522

## Geometrical Methods of Mathematical Physics

**Author**: Bernard F. Schutz**Publisher:**Cambridge University Press**ISBN:**1107268141**Category:**Science**Page:**N.A**View:**5261

In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions.

## The Geometry of Physics

*An Introduction*

**Author**: Theodore Frankel**Publisher:**Cambridge University Press**ISBN:**1139505610**Category:**Mathematics**Page:**N.A**View:**990

This book provides a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism (in flat and curved space), thermodynamics, the Dirac operator and spinors, and gauge fields, including Yang–Mills, the Aharonov–Bohm effect, Berry phase and instanton winding numbers, quarks and quark model for mesons. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. The book is ideal for graduate and advanced undergraduate students of physics, engineering or mathematics as a course text or for self study. This third edition includes an overview of Cartan's exterior differential forms, which previews many of the geometric concepts developed in the text.

## Differentiable Manifolds

*A Theoretical Physics Approach*

**Author**: Gerardo F. Torres del Castillo**Publisher:**Springer Science & Business Media**ISBN:**9780817682712**Category:**Mathematics**Page:**275**View:**2988

This textbook delves into the theory behind differentiable manifolds while exploring various physics applications along the way. Included throughout the book are a collection of exercises of varying degrees of difficulty. Differentiable Manifolds is intended for graduate students and researchers interested in a theoretical physics approach to the subject. Prerequisites include multivariable calculus, linear algebra, and differential equations and a basic knowledge of analytical mechanics.

## Differential Geometry and Mathematical Physics

*Part II. Fibre Bundles, Topology and Gauge Fields*

**Author**: Gerd Rudolph,Matthias Schmidt**Publisher:**Springer**ISBN:**9402409599**Category:**Science**Page:**830**View:**4664

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.

## Curvature in Mathematics and Physics

**Author**: Shlomo Sternberg**Publisher:**Courier Corporation**ISBN:**0486292711**Category:**Mathematics**Page:**416**View:**4680

Expert treatment introduces semi-Riemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool. Prerequisites include linear algebra and advanced calculus. 2012 edition.

## Geometric and Topological Methods for Quantum Field Theory

**Author**: Hernan Ocampo,Eddy Pariguan,Sylvie Paycha**Publisher:**Cambridge University Press**ISBN:**113948673X**Category:**Science**Page:**N.A**View:**2572

Aimed at graduate students in physics and mathematics, this book provides an introduction to recent developments in several active topics at the interface between algebra, geometry, topology and quantum field theory. The first part of the book begins with an account of important results in geometric topology. It investigates the differential equation aspects of quantum cohomology, before moving on to noncommutative geometry. This is followed by a further exploration of quantum field theory and gauge theory, describing AdS/CFT correspondence, and the functional renormalization group approach to quantum gravity. The second part covers a wide spectrum of topics on the borderline of mathematics and physics, ranging from orbifolds to quantum indistinguishability and involving a manifold of mathematical tools borrowed from geometry, algebra and analysis. Each chapter presents introductory material before moving on to more advanced results. The chapters are self-contained and can be read independently of the rest.

## Riemann, Topology, and Physics

**Author**: Michael Monastyrsky**Publisher:**Springer Science & Business Media**ISBN:**1489935142**Category:**Mathematics**Page:**158**View:**4215

Soviet citizens can buy Monastyrsky's biography of Riemann for eleven kopeks. This translated edition will cost considerably more, but it is still good value for the money. And we get Monastyrsky's monograph on topological methods in the bargain. It was a good idea of Birkhiiuser Boston to publish the two translations in one volume. The economics of publishing in a capitalist country make it impossible for us to produce the small cheap paperback booklets, low in quality of paper and high in quality of scholarship, at which the Soviet publishing industry excels. Monastyrsky's two booklets are out standing examples of the genre. By putting them together, Birkhiiuser has enabled them to fit into the Western book-marketing system. The two booklets were written separately and each is complete in itself, but they complement each other beautifully. The Riemann biography is short and terse, like Riemann's own writings. It describes in few words and fewer equations the revolutionary ideas which Riemann brought into mathematics and physics a hundred and twenty years ago. The topological methods booklet describes how some of these same ideas, after lying dormant for a century, found new and fruitful applications in the physics of our own time.

## Applied Differential Geometry

**Author**: William L. Burke**Publisher:**Cambridge University Press**ISBN:**9780521269292**Category:**Mathematics**Page:**414**View:**3233

This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples.

## Differential Forms in Mathematical Physics

**Author**: N.A**Publisher:**Elsevier**ISBN:**9780080875248**Category:**Mathematics**Page:**484**View:**7392

Differential Forms in Mathematical Physics

## Global Analysis

*Differential Forms in Analysis, Geometry, and Physics*

**Author**: Ilka Agricola,Thomas Friedrich**Publisher:**American Mathematical Soc.**ISBN:**0821829513**Category:**Mathematics**Page:**343**View:**9879

This book introduces the reader to the world of differential forms and their uses in geometry, analysis, and mathematical physics. It begins with a few basic topics, partly as review, then moves on to vector analysis on manifolds and the study of curves and surfaces in $3$-space. Lie groups and homogeneous spaces are discussed, providing the appropriate framework for introducing symmetry in both mathematical and physical contexts. The final third of the book applies the mathematical ideas to important areas of physics: Hamiltonian mechanics, statistical mechanics, and electrodynamics. There are many classroom-tested exercises and examples with excellent figures throughout. The book is ideal as a text for a first course in differential geometry, suitable for advanced undergraduates or graduate students in mathematics or physics.

## Applied Differential Geometry

*A Modern Introduction*

**Author**: Vladimir G. Ivancevic,Tijana T. Ivancevic**Publisher:**World Scientific**ISBN:**9812706143**Category:**Mathematics**Page:**1311**View:**5556

This graduate-level monographic textbook treats applied differential geometry from a modern scientific perspective. Co-authored by the originator of the world's leading human motion simulator ? ?Human Biodynamics Engine?, a complex, 264-DOF bio-mechanical system, modeled by differential-geometric tools ? this is the first book that combines modern differential geometry with a wide spectrum of applications, from modern mechanics and physics, via nonlinear control, to biology and human sciences. The book is designed for a two-semester course, which gives mathematicians a variety of applications for their theory and physicists, as well as other scientists and engineers, a strong theory underlying their models.

## An Introduction to Symplectic Geometry

**Author**: Rolf Berndt**Publisher:**American Mathematical Soc.**ISBN:**9780821820568**Category:**Mathematics**Page:**195**View:**2318

Starts with the basics of the geometry of symplectic vector spaces. Then, symplectic manifolds are defined and explored. In addition to the essential classic results, such as Darboux's theorem, more recent results and ideas are also included here, such as symplectic capacity and pseudoholomorphic curves. These ideas have revolutionized the subject. The main examples of symplectic manifolds are given, including the cotangent bundle, Kahler manifolds, and coadjoint orbits.Further principal ideas are carefully examined, such as Hamiltonian vector fields, the Poisson bracket, and connections with contact manifolds. Berndt describes some of the close connections between symplectic geometry and mathematical physics in the last two chapters of the book. In particular, the moment map is defined and explored, both mathematically and in its relation to physics. He also introduces symplectic reduction, which is an important tool for reducing the number of variables in a physical system and for constructing new symplectic manifolds from old. The final chapter is on quantization, which uses symplectic methods to take classical mechanics to quantum mechanics.This section includes a discussion of the Heisenberg group and the Weil (or metaplectic) representation of the symplectic group. Several appendices provide background material on vector bundles, on cohomology, and on Lie groups and Lie algebras and their representations.