# 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:**6324

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.

## Quaternionic Structures in Mathematics and Physics

*Proceedings of the Second Meeting : Rome, Italy, 6-10 September 1999*

**Author**: Stefano Marchiafava,Paolo Piccinni,Massimiliano Pontecorvo**Publisher:**World Scientific**ISBN:**981281003X**Category:**Mathematics**Page:**469**View:**6054

During the last five years, after the first meeting on OC Quaternionic Structures in Mathematics and PhysicsOCO, interest in quaternionic geometry and its applications has continued to increase. Progress has been made in constructing new classes of manifolds with quaternionic structures (quaternionic Knhler, hyper-Knhler, hyper-complex, etc.), studying the differential geometry of special classes of such manifolds and their submanifolds, understanding relations between the quaternionic structure and other differential-geometric structures, and also in physical applications of quaternionic geometry. Some generalizations of classical quaternion-like structures (like HKT structures and hyper-Knhler manifolds with singularities) appeared naturally and were studied. Some of those results are published in this book. Contents: Hypercomplex Structures on Special Classes of Nilpotent and Solvable Lie Groups (M L Barberis); Twistor Quotients of HyperKnhler Manifolds (R Bielawski); Quaternionic Contact Structures (O Biquard); A New Construction of Homogeneous Quaternionic Manifolds and Related Geometric Structures (V Cortes); Quaternion Knhler Flat Manifolds (I G Dotti); A Canonical HyperKnhler Metric on the Total Space of a Cotangent Bundle (D Kaledin); Special Spinors and Contact Geometry (A Moroianu); Brane Solitons and Hypercomplex Structures (G Papadopoulos); Hypercomplex Geometry (H Pedersen); Examples of HyperKnhler Connections with Torsion (Y S Poon); A New Weight System on Chord Diagrams via HyperKnhler Geometry (J Sawon); Vanishing Theorems for Quaternionic Knhler Manifolds (U Semmelmann & G Weingart); Weakening Holonomy (A Swann); Special Knhler Geometry (A Van Proeyen); Singularities in HyperKnhler Geometry (M Verbitsky); and other papers. Readership: Researchers and graduate students in geometry, topology, mathematical physics and theoretical physics."

## Einführung in die Geometrie und Topologie

**Author**: Werner Ballmann**Publisher:**Springer-Verlag**ISBN:**3034809018**Category:**Mathematics**Page:**162**View:**1483

Das Buch bietet eine Einführung in die Topologie, Differentialtopologie und Differentialgeometrie. Es basiert auf Manuskripten, die in verschiedenen Vorlesungszyklen erprobt wurden. Im ersten Kapitel werden grundlegende Begriffe und Resultate aus der mengentheoretischen Topologie bereitgestellt. Eine Ausnahme hiervon bildet der Jordansche Kurvensatz, der für Polygonzüge bewiesen wird und eine erste Idee davon vermitteln soll, welcher Art tiefere topologische Probleme sind. Im zweiten Kapitel werden Mannigfaltigkeiten und Liesche Gruppen eingeführt und an einer Reihe von Beispielen veranschaulicht. Diskutiert werden auch Tangential- und Vektorraumbündel, Differentiale, Vektorfelder und Liesche Klammern von Vektorfeldern. Weiter vertieft wird diese Diskussion im dritten Kapitel, in dem die de Rhamsche Kohomologie und das orientierte Integral eingeführt und der Brouwersche Fixpunktsatz, der Jordan-Brouwersche Zerlegungssatz und die Integralformel von Stokes bewiesen werden. Das abschließende vierte Kapitel ist den Grundlagen der Differentialgeometrie gewidmet. Entlang der Entwicklungslinien, die die Geometrie der Kurven und Untermannigfaltigkeiten in Euklidischen Räumen durchlaufen hat, werden Zusammenhänge und Krümmung, die zentralen Konzepte der Differentialgeometrie, diskutiert. Den Höhepunkt bilden die Gaussgleichungen, die Version des theorema egregium von Gauss für Untermannigfaltigkeiten beliebiger Dimension und Kodimension. Das Buch richtet sich in erster Linie an Mathematik- und Physikstudenten im zweiten und dritten Studienjahr und ist als Vorlage für ein- oder zweisemestrige Vorlesungen geeignet.

## Operads in Algebra, Topology and Physics

**Author**: Martin Markl,Steven Shnider,James D. Stasheff**Publisher:**American Mathematical Soc.**ISBN:**0821843621**Category:**Mathematics**Page:**349**View:**6814

'Operads are powerful tools, and this is the book in which to read about them' - ""Bulletin of the London Mathematical Society"". Operads are mathematical devices that describe algebraic structures of many varieties and in various categories. Operads are particularly important in categories with a good notion of 'homotopy', where they play a key role in organizing hierarchies of higher homotopies. Significant examples from algebraic topology first appeared in the sixties, although the formal definition and appropriate generality were not forged until the seventies. In the nineties, a renaissance and further development of the theory were inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, derived categories, Morse theory, symplectic and contact geometry, combinatorics, knot theory, moduli spaces, cyclic cohomology, and, last but not least, theoretical physics, especially string field theory and deformation quantization. The book contains a detailed and comprehensive historical introduction describing the development of operad theory from the initial period when it was a rather specialized tool in homotopy theory to the present when operads have a wide range of applications in algebra, topology, and mathematical physics. Many results and applications currently scattered in the literature are brought together here along with new results and insights. The basic definitions and constructions are carefully explained and include many details not found in any of the standard literature.

## Differential Geometric Methods in Theoretical Physics

*Physics and Geometry*

**Author**: Ling-Lie Chau,Werner Nahm**Publisher:**Springer Science & Business Media**ISBN:**1468491482**Category:**Mathematics**Page:**830**View:**885

After several decades of reduced contact, the interaction between physicists and mathematicians in the front-line research of both fields recently became deep and fruit ful again. Many of the leading specialists of both fields became involved in this devel opment. This process even led to the discovery of previously unsuspected connections between various subfields of physics and mathematics. In mathematics this concerns in particular knots von Neumann algebras, Kac-Moody algebras, integrable non-linear partial differential equations, and differential geometry in low dimensions, most im portantly in three and four dimensional spaces. In physics it concerns gravity, string theory, integrable classical and quantum field theories, solitons and the statistical me chanics of surfaces. New discoveries in these fields are made at a rapid pace. This conference brought together active researchers in these areas, reporting their results and discussing with other participants to further develop thoughts in future new directions. The conference was attended by SO participants from 15 nations. These proceedings document the program and the talks at the conference. This conference was preceded by a two-week summer school. Ten lecturers gave extended lectures on related topics. The proceedings of the school will also be published in the NATO-AS[ volume by Plenum. The Editors vii ACKNOWLEDGMENTS We would like to thank the many people who have made the conference a success. Furthermore, ·we appreciate the excellent talks. The active participation of everyone present made the conference lively and stimulating. All of this made our efforts worth while.

## An Introduction to Symplectic Geometry

**Author**: Rolf Berndt**Publisher:**American Mathematical Soc.**ISBN:**9780821820568**Category:**Mathematics**Page:**195**View:**5334

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.

## Topics in Statistical and Theoretical Physics

*F.A. Berezin Memorial Volume*

**Author**: R. L. Dobrushin**Publisher:**American Mathematical Soc.**ISBN:**9780821804254**Category:**Mathematical physics**Page:**223**View:**578

This is the second of two volumes dedicated to the scientific heritage of F. A. Berezin (1931-1980). Before his untimely death, Berezin had an important influence on physics and mathematics, discovering new ideas in mathematical physics, representation theory, analysis, geometry, and other areas of mathematics. His crowning achievements were the introduction of a new notion of deformation quantization and Grassmannian analysis (supermathematics). Collected here are papers by many of his colleagues and others who worked in related areas, representing a wide spectrum of topics in statistical and theoretical physics and allied areas of mathematics. In particular, several papers discuss various aspects of quantum field theory and related questions of supersymmetry, geometry, and representation theory. Other papers are devoted to problems of quasi-classical approximation and mathematical models of statistical physics.

## Visions in Mathematics

*GAFA 2000 Special Volume, Part II*

**Author**: Noga Alon,Jean Bourgain,Alain Connes,Misha Gromov,Vitali D. Milman**Publisher:**Springer Science & Business Media**ISBN:**9783034604253**Category:**Mathematics**Page:**528**View:**624

"Visions in Mathematics - Towards 2000" was one of the most remarkable mathematical meetings in recent years. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas. The aims of the conference are reflected in the present set of survey articles, documenting the state of art and future prospects in many branches of mathematics of current interest. This is the second part of a two-volume set that will serve any research mathematician or advanced student as an overview and guideline through the multifaceted body of mathematical research in the present and near future.

## Geometry And Topology Of Submanifolds Viii

**Author**: Van De Woestyne Ignace,Dillen Franki,Simon Udo**Publisher:**World Scientific**ISBN:**9814547514**Category:****Page:**424**View:**3474

This is a volume in honor of Professor Peter Carruthers on the occasion of his 61st birthday. It is a unique collection of papers by the world's leading experts, describing the most exciting developments in many areas of theoretical physics. While traditionally physics is driven to ever smaller and simpler systems, end-of-this-century scientists see themselves confronted with complex systems in many of their areas. It is just this interdisciplinary character of complexity that is addressed in this book, with topics ranging from the origin of intelligent life and of universal scaling laws in biology via heartbeats, proteins, fireballs, phase transitions, all the way to parton branching in collisions of elementary particles at high energies. The contributions include extensive discussions on complexity (M Gell-Mann, M Feigenbaum, D Champbell, D Pines and L M Simmons), neutrino masses (R Slansky and P Rosen), high temperature superconductors (D Pines), low Moon (M Feigenbaum), origin of intelligent life (S Colgate), chaos of the heart (M Duong-Van), origin of universal scaling laws in biological systems (G West), critical behavior of quarks (R Hwa), status of LEGO (S Meshov), disoriented chiral condensate (F Cooper), and many others.

## Applied Differential Geometry

**Author**: William L. Burke**Publisher:**Cambridge University Press**ISBN:**9780521269292**Category:**Mathematics**Page:**414**View:**419

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 Geometry for Physicists

**Author**: Bo-Yu Hou,Bo-Yuan Hou**Publisher:**World Scientific Publishing Company**ISBN:**9813105097**Category:**Mathematics**Page:**560**View:**837

This book is divided into fourteen chapters, with 18 appendices as introduction to prerequisite topological and algebraic knowledge, etc. The first seven chapters focus on local analysis. This part can be used as a fundamental textbook for graduate students of theoretical physics. Chapters 8–10 discuss geometry on fibre bundles, which facilitates further reference for researchers. The last four chapters deal with the Atiyah-Singer index theorem, its generalization and its application, quantum anomaly, cohomology field theory and noncommutative geometry, giving the reader a glimpse of the frontier of current research in theoretical physics.

## Geometry and Physics

**Author**: H. Pedersen,J. Andersen,J. Dupont,Andrew Swann**Publisher:**CRC Press**ISBN:**9780824797911**Category:**Mathematics**Page:**768**View:**7214

"Based on the proceedings of the Special Session on Geometry and Physics held over a six month period at the University of Aarhus, Denmark and on articles from the Summer school held at Odense University, Denmark. Offers new contributions on a host of topics that involve physics, geometry, and topology. Written by more than 50 leading international experts."

## Differential Forms with Applications to the Physical Sciences

**Author**: Harley Flanders**Publisher:**Courier Corporation**ISBN:**9780486661698**Category:**Mathematics**Page:**205**View:**2314

Introduces the use of exterior differential forms as a powerful took in the analysis of a variety of mathematical problems in the physical and engineering sciences.

## Singularities of Caustics and Wave Fronts

**Author**: Vladimir Arnold**Publisher:**Springer Science & Business Media**ISBN:**9401133301**Category:**Mathematics**Page:**259**View:**1332

One service mathematics has rendered the 'Et moi ...) si j'avait su comment en revenir, human race. It has put common sense back je n'y serais point aile.' Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we may be sense'. ErieT. Bell able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

## Contact Geometry and Nonlinear Differential Equations

**Author**: Alexei Kushner,Valentin Lychagin,Vladimir Rubtsov**Publisher:**Cambridge University Press**ISBN:**0521824761**Category:**Mathematics**Page:**496**View:**7795

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

## Mathematical Methods of Classical Mechanics

**Author**: V.I. Arnol'd**Publisher:**Springer Science & Business Media**ISBN:**1475720637**Category:**Mathematics**Page:**520**View:**1182

This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.

## Dynamical Systems IV

*Symplectic Geometry and its Applications*

**Author**: V.I. Arnol'd,S.P. Novikov**Publisher:**Springer Science & Business Media**ISBN:**3662067935**Category:**Mathematics**Page:**286**View:**9927

This book takes a snapshot of the mathematical foundations of classical and quantum mechanics from a contemporary mathematical viewpoint. It covers a number of important recent developments in dynamical systems and mathematical physics and places them in the framework of the more classical approaches; the presentation is enhanced by many illustrative examples concerning topics which have been of especial interest to workers in the field, and by sketches of the proofs of the major results. The comprehensive bibliographies are designed to permit the interested reader to retrace the major stages in the development of the field if he wishes. Not so much a detailed textbook for plodding students, this volume, like the others in the series, is intended to lead researchers in other fields and advanced students quickly to an understanding of the 'state of the art' in this area of mathematics. As such it will serve both as a basic reference work on important areas of mathematical physics as they stand today, and as a good starting point for further, more detailed study for people new to this field.

## Protein Geometry, Classification, Topology and Symmetry

*A Computational Analysis of Structure*

**Author**: William R. Taylor,Andras Aszodi**Publisher:**CRC Press**ISBN:**1420033638**Category:**Science**Page:**348**View:**669

Using a geometric perspective, Protein Geometry, Classification, Topology, and Symmetry reviews and analyzes the structural principals of proteins with the goal of revealing the underlying regularities in their construction. It also reviews computer methods for structure analysis and the automatic comparison and classification of these structures with an analysis of the statistical significance of comparing different shapes. Following an analysis of the current state of protein classification, the authors explore more abstract geometric and topological representations, including the occurrence of knotted topologies. The book concludes with a consideration of the origin of higher-level symmetries in protein structure. The authors focus on simple geometric methods that are deterministic rather than probabilistic and on the more abstract simplifications of protein structure that allow a better understanding of the overall fold of the structure. Most of the methods described in this book have corresponding computer programs. These can be found (as C source code) at the ftp site of the Division of Mathematical Biology at the National Institute for Medical Research. This collection of ideas contains pedagogical material that make it ideal for post-graduate courses as well as new ideas and results essential for researchers investigating protein structures.

## The Theory of Singularities and Its Applications

**Author**: V. I. Arnold,Vladimir Igorevich Arnolʹd**Publisher:**Cambridge University Press**ISBN:**9780521422802**Category:**Mathematics**Page:**72**View:**4296

This book describes those singularities encountered in different branches of mathematics. The distinguished mathematician, Vladimir Arnold, avoids giving difficult proofs of all the results in order to provide the reader with a concise and accessible overview of the many guises and areas in which singularities appear. Some of these areas include geometry and optics, optimal control theory and algebraic geometry, reflection groups theory, dynamical systems theory, and the classical and quantum catastrophe theory.

## Introduction to Smooth Manifolds

**Author**: John Lee**Publisher:**Springer Science & Business Media**ISBN:**1441999825**Category:**Mathematics**Page:**708**View:**6673

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer. This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures. Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.