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

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: 1680
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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.

Differentialgeometrie, Topologie und Physik

Differentialgeometrie, Topologie und Physik

  • Author: Mikio Nakahara
  • Publisher: Springer-Verlag
  • ISBN: 3662453002
  • Category: Science
  • Page: 597
  • View: 7817
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Differentialgeometrie und Topologie sind wichtige Werkzeuge für die Theoretische Physik. Insbesondere finden sie Anwendung in den Gebieten der Astrophysik, der Teilchen- und Festkörperphysik. Das vorliegende beliebte Buch, das nun erstmals ins Deutsche übersetzt wurde, ist eine ideale Einführung für Masterstudenten und Forscher im Bereich der theoretischen und mathematischen Physik. - Im ersten Kapitel bietet das Buch einen Überblick über die Pfadintegralmethode und Eichtheorien. - Kapitel 2 beschäftigt sich mit den mathematischen Grundlagen von Abbildungen, Vektorräumen und der Topologie. - Die folgenden Kapitel beschäftigen sich mit fortgeschritteneren Konzepten der Geometrie und Topologie und diskutieren auch deren Anwendungen im Bereich der Flüssigkristalle, bei suprafluidem Helium, in der ART und der bosonischen Stringtheorie. - Daran anschließend findet eine Zusammenführung von Geometrie und Topologie statt: es geht um Faserbündel, characteristische Klassen und Indextheoreme (u.a. in Anwendung auf die supersymmetrische Quantenmechanik). - Die letzten beiden Kapitel widmen sich der spannendsten Anwendung von Geometrie und Topologie in der modernen Physik, nämlich den Eichfeldtheorien und der Analyse der Polakov'schen bosonischen Stringtheorie aus einer gemetrischen Perspektive. Mikio Nakahara studierte an der Universität Kyoto und am King’s in London Physik sowie klassische und Quantengravitationstheorie. Heute ist er Physikprofessor an der Kinki-Universität in Osaka (Japan), wo er u. a. über topologische Quantencomputer forscht. Diese Buch entstand aus einer Vorlesung, die er während Forschungsaufenthalten an der University of Sussex und an der Helsinki University of Sussex gehalten hat.

Quaternionic Structures in Mathematics and Physics

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: 4874
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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."

Operads in Algebra, Topology and Physics

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: 9787
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'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.

Geometry And Topology Of Submanifolds Viii

Geometry And Topology Of Submanifolds Viii

  • Author: Van De Woestyne Ignace,Dillen Franki,Simon Udo
  • Publisher: World Scientific
  • ISBN: 9814547514
  • Category:
  • Page: 424
  • View: 321
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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.

Direkte Methoden der Variationsrechnung

Direkte Methoden der Variationsrechnung

Ein Lehrbuch

  • Author: Ph. Blanchard,E. Brüning
  • Publisher: Springer-Verlag
  • ISBN: 3709122600
  • Category: Science
  • Page: 280
  • View: 5193
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Geometry, Topology, and Mathematical Physics

Geometry, Topology, and Mathematical Physics

S.P. Novikov's Seminar, 2002-2003

  • Author: V. M. Buchstaber,Sergeĭ Petrovich Novikov,I. M. Krichever
  • Publisher: American Mathematical Soc.
  • ISBN: 9780821836132
  • Category: Mathematics
  • Page: 324
  • View: 5078
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This volume contains a selection of papers based on presentations given at the S. P. Novikov seminar held at the Steklov Mathematical Institute in Moscow. The topics and speakers were chosen by the well-known expert, S. P. Novikov, one of the leading mathematicians of the twentieth century. His diverse interests are the tradition of the seminar and are reflected in the topics presented in the book. The book begins with Novikov's paper analyzing the position of mathematics and theoretical physics at the beginning of the new millennium. Following is an interview with Novikov published in the ""Newsletter of the European Mathematical Society"" presenting the genesis of many of his ideas and his scientific school. The remaining articles address topics in geometry, topology, and mathematical physics. The volume is suitable for graduate students and researchers interested in the corresponding areas of mathematics and physics.

Einführung in die Geometrie und Topologie

Einführung in die Geometrie und Topologie

  • Author: Werner Ballmann
  • Publisher: Springer-Verlag
  • ISBN: 3034809018
  • Category: Mathematics
  • Page: 162
  • View: 7865
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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.

An Introduction to Symplectic Geometry

An Introduction to Symplectic Geometry

  • Author: Rolf Berndt
  • Publisher: American Mathematical Soc.
  • ISBN: 9780821820568
  • Category: Mathematics
  • Page: 195
  • View: 7195
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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.

Singularities of Caustics and Wave Fronts

Singularities of Caustics and Wave Fronts

  • Author: Vladimir Arnold
  • Publisher: Springer Science & Business Media
  • ISBN: 9401133301
  • Category: Mathematics
  • Page: 259
  • View: 794
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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.

Einführung in die Symplektische Geometrie

Einführung in die Symplektische Geometrie

  • Author: Rolf Berndt
  • Publisher: Springer-Verlag
  • ISBN: 9783322802156
  • Category: Mathematics
  • Page: 185
  • View: 6688
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Differential Geometry for Physicists

Differential Geometry for Physicists

  • Author: Bo-Yu Hou,Bo-Yuan Hou
  • Publisher: World Scientific Publishing Company
  • ISBN: 9813105097
  • Category: Mathematics
  • Page: 560
  • View: 654
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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.

Applied Differential Geometry

Applied Differential Geometry

  • Author: William L. Burke
  • Publisher: Cambridge University Press
  • ISBN: 9780521269292
  • Category: Mathematics
  • Page: 414
  • View: 1214
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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.

Contact Geometry and Nonlinear Differential Equations

Contact Geometry and Nonlinear Differential Equations

  • Author: Alexei Kushner,Valentin Lychagin,Vladimir Rubtsov
  • Publisher: Cambridge University Press
  • ISBN: 0521824761
  • Category: Mathematics
  • Page: 496
  • View: 5748
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Shows novel and modern ways of solving differential equations using methods from contact and symplectic geometry.

Topics in Statistical and Theoretical Physics

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: 8683
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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

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: 4928
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"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.

Differential Forms with Applications to the Physical Sciences

Differential Forms with Applications to the Physical Sciences

  • Author: Harley Flanders
  • Publisher: Courier Corporation
  • ISBN: 9780486661698
  • Category: Mathematics
  • Page: 205
  • View: 7166
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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.

Distributionen Und Hilbertraumoperatoren

Distributionen Und Hilbertraumoperatoren

Mathematische Methoden Der Physik

  • Author: Philippe Blanchard,Erwin Brüning
  • Publisher: Springer
  • ISBN: 9783211825075
  • Category: Science
  • Page: 375
  • View: 3311
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Das Buch bietet eine Einführung in die zum Studium der Theoretischen Physik notwendigen mathematischen Grundlagen. Der erste Teil des Buches beschäftigt sich mit der Theorie der Distributionen und vermittelt daneben einige Grundbegriffe der linearen Funktionalanalysis. Der zweite Teil baut darauf auf und gibt eine auf das Wesentliche beschränkte Einführung in die Theorie der linearen Operatoren in Hilbert-Räumen. Beide Teile werden von je einer Übersicht begleitet, die die zentralen Ideen und Begriffe knapp erläutert und den Inhalt kurz beschreibt. In den Anhängen werden einige grundlegende Konstruktionen und Konzepte der Funktionalanalysis dargestellt und wichtige Konsequenzen entwickelt.

Raum, Zeit, Materie

Raum, Zeit, Materie

  • Author: Hermann Weyl
  • Publisher: Рипол Классик
  • ISBN: 5880098028
  • Category: History
  • Page: N.A
  • View: 8068
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Morphology of Condensed Matter

Morphology of Condensed Matter

Physics and Geometry of Spatially Complex Systems

  • Author: Klaus R. Mecke,Dietrich Stoyan
  • Publisher: Springer
  • ISBN: 3540457828
  • Category: Science
  • Page: 442
  • View: 5667
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The morphology of spatially stuctured materials is a rapidly growing field of research at the interface of statistical physics, applied mathematics and materials science. A wide spectrum of applications encompasses the flow through porous and composite materials as well as microemulsions and foams. Written as a set of lectures and tutorial reviews leading up to the forefront of research, this book will be both a compendium for the experienced researcher as well as a high level introductory text for postgraduate students and nonspecialist researchers working in related areas.