Applications of Contact Geometry and Topology in Physics

Applications of Contact Geometry and Topology in Physics

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.

Author: Arkady Leonidovich Kholodenko

Publisher: World Scientific

ISBN: 9789814412094

Category: Mathematics

Page: 492

View: 608

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.
Categories: Mathematics

Quaternionic Structures in Mathematics and Physics

Quaternionic Structures in Mathematics and Physics

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.

Author: Stefano Marchiafava

Publisher: World Scientific

ISBN: 9789812810038

Category: Mathematics

Page: 469

View: 263

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."
Categories: Mathematics

Contact Geometry and Nonlinear Differential Equations

Contact Geometry and Nonlinear Differential Equations

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

Author: Alexei Kushner

Publisher: Cambridge University Press

ISBN: 9780521824767

Category: Mathematics

Page: 496

View: 843

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

Operads in Algebra Topology and Physics

Operads in Algebra  Topology and Physics

'Operads are powerful tools, and this is the book in which to read about them' - ""Bulletin of the London Mathematical Society"".

Author: Martin Markl

Publisher: American Mathematical Soc.

ISBN: 9780821843628

Category: Mathematics

Page: 349

View: 290

'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.
Categories: Mathematics

Hyperspatial Dynamics

Hyperspatial Dynamics

... geometry, contact geometry has broad applications in physics, e.g. geometrical
optics, classical mechanics, thermodynamics, geometric quantization, and to
control theory. Contact geometry also has applications to low-dimensional
topology ...

Author: Dr. Marco Bitetto

Publisher: Dr. Marco A. V. Bitetto

ISBN:

Category: Architecture

Page: 503

View: 510

This dissertation has as its central focus the study of hyperspatial dynamics and as such makes use of mathematics in such an understanding and also the MAXYMA artificial intelligence computer simulation and programming language. As such, it will both discuss the use of MAXYMA in the understanding of hyperspatial dynamics and also include MAXYMA programs as well. This dissertation will conclude with a discussion of hyperspace and how one can travel through hyperspace and why one would want to travel through hyperspace.
Categories: Architecture

Visions in Mathematics

Visions in Mathematics

The contact-geometric ingredient of our work is greatly motivated by two
outstanding conjectures in contact geometry: Weinstein's conjecture about
periodic orbits of Reeb fields [W], and ... Presently, we are working on a series of
papers devoted to the foundations, applications, and further development of SFT.
Among ... We are expecting new links with the low-dimensional topology and,
possibly, Physics.

Author: Noga Alon

Publisher: Springer Science & Business Media

ISBN: 3034604254

Category: Mathematics

Page: 528

View: 258

"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.
Categories: Mathematics

Aspects Of Complex Analysis Differential Geometry Mathematical Physics And Applications Proceedings Of The Fourth International Workshop On Complex Structures And Vector Fields

Aspects Of Complex Analysis  Differential Geometry  Mathematical Physics And Applications   Proceedings Of The Fourth International Workshop On Complex Structures And Vector Fields

This volume constitutes the proceedings of a workshop whose main purpose was to exchange information on current topics in complex analysis, differential geometry, mathematical physics and applications, and to group aspects of new ...

Author: Dimiev Stancho

Publisher: World Scientific

ISBN: 9789814543750

Category: Mathematics

Page: 380

View: 549

This volume constitutes the proceedings of a workshop whose main purpose was to exchange information on current topics in complex analysis, differential geometry, mathematical physics and applications, and to group aspects of new mathematics.
Categories: Mathematics

Computational Geometry Topology and Physics of Digital Images with Applications

Computational Geometry  Topology and Physics of Digital Images with Applications

This Springer imprint is published by the registered company Springer Nature
Switzerland AG The registered company address is: Gewerbestrasse 11, 6330
Cham, Switzerland This book is dedicated to Somashekhar (Som) Naimpally, ...

Author: James F. Peters

Publisher: Springer Nature

ISBN: 9783030221928

Category: Technology & Engineering

Page: 440

View: 321

This book discusses the computational geometry, topology and physics of digital images and video frame sequences. This trio of computational approaches encompasses the study of shape complexes, optical vortex nerves and proximities embedded in triangulated video frames and single images, while computational geometry focuses on the geometric structures that infuse triangulated visual scenes. The book first addresses the topology of cellular complexes to provide a basis for an introductory study of the computational topology of visual scenes, exploring the fabric, shapes and structures typically found in visual scenes. The book then examines the inherent geometry and topology of visual scenes, and the fine structure of light and light caustics of visual scenes, which bring into play catastrophe theory and the appearance of light caustic folds and cusps. Following on from this, the book introduces optical vortex nerves in triangulated digital images. In this context, computational physics is synonymous with the study of the fine structure of light choreographed in video frames. This choreography appears as a sequence of snapshots of light reflected and refracted from surface shapes, providing a solid foundation for detecting, analyzing and classifying visual scene shapes.
Categories: Technology & Engineering

Topics in Statistical and Theoretical Physics

Topics in Statistical and Theoretical Physics

ITT, 1996 Superanalogs of Symplectic and Contact Geometry and Their
Applications to Quantum Field Theory Albert ... geometry, one can better
understand N = 2 superconformal field theory and its relationship to topological
conformal field ...

Author: R. L. Dobrushin

Publisher: American Mathematical Soc.

ISBN: 0821804251

Category: Mathematical physics

Page: 223

View: 293

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.
Categories: Mathematical physics

Dynamical Systems IV

Dynamical Systems IV

This book takes a snapshot of the mathematical foundations of classical and quantum mechanics from a contemporary mathematical viewpoint.

Author: V.I. Arnol'd

Publisher: Springer Science & Business Media

ISBN: 9783662067932

Category: Mathematics

Page: 286

View: 629

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.
Categories: Mathematics

An Introduction to Symplectic Geometry

An Introduction to Symplectic Geometry

This book would be an excellent text for a graduate course or as a source for anyone who wishes to learn about symplectic geometry.

Author: Rolf Berndt

Publisher: American Mathematical Soc.

ISBN: 0821820567

Category: Mathematics

Page: 195

View: 537

Symplectic geometry is a central topic of current research in mathematics. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of Lie groups. This book is a true introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra. It 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. Berndt's presentation of symplectic geometry is a clear and concise introduction to the major methods and applications of the subject, and requires only a minimum of prerequisites. This book would be an excellent text for a graduate course or as a source for anyone who wishes to learn about symplectic geometry.
Categories: Mathematics

Differential Equations Geometry Symmetries and Integrability

Differential Equations   Geometry  Symmetries and Integrability

This volume consists of original contributions and broad overview lectures of the participants of the Symposium. The papers in this volume present the modern approach to this classical subject.

Author: Boris Kruglikov

Publisher: Springer Science & Business Media

ISBN: 9783642008733

Category: Mathematics

Page: 386

View: 249

The Abel Symposium 2008 focused on the modern theory of differential equations and their applications in geometry, mechanics, and mathematical physics. Following the tradition of Monge, Abel and Lie, the scientific program emphasized the role of algebro-geometric methods, which nowadays permeate all mathematical models in natural and engineering sciences. The ideas of invariance and symmetry are of fundamental importance in the geometric approach to differential equations, with a serious impact coming from the area of integrable systems and field theories. This volume consists of original contributions and broad overview lectures of the participants of the Symposium. The papers in this volume present the modern approach to this classical subject.
Categories: Mathematics

Applied Differential Geometry

Applied Differential Geometry

This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry.

Author: William L. Burke

Publisher: Cambridge University Press

ISBN: 0521269296

Category: Mathematics

Page: 414

View: 986

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.
Categories: Mathematics

Floer Homology Gauge Theory and Low Dimensional Topology

Floer Homology  Gauge Theory  and Low Dimensional Topology

This volume is based on lecture courses and advanced seminars given at the 2004 Clay Mathematics Institute Summer School at the Alfred Renyi Institute of Mathematics in Budapest, Hungary.

Author: Clay Mathematics Institute. Summer School

Publisher: American Mathematical Soc.

ISBN: 0821838458

Category: Mathematics

Page: 297

View: 288

Mathematical gauge theory studies connections on principal bundles, or, more precisely, the solution spaces of certain partial differential equations for such connections. Historically, these equations have come from mathematical physics, and play an important role in the description of the electro-weak and strong nuclear forces. The use of gauge theory as a tool for studying topological properties of four-manifolds was pioneered by the fundamental work of Simon Donaldson in the early 1980s, and was revolutionized by the introduction of the Seiberg-Witten equations in the mid-1990s. Since the birth of the subject, it has retained its close connection with symplectic topology. The analogy between these two fields of study was further underscored by Andreas Floer's construction of an infinite-dimensional variant of Morse theory that applies in two a priori different contexts: either to define symplectic invariants for pairs of Lagrangian submanifolds of a symplectic manifold, or to define topological invariants for three-manifolds, which fit into a framework for calculating invariants for smooth four-manifolds. ``Heegaard Floer homology'', the recently-discovered invariant for three- and four-manifolds, comes from an application of Lagrangian Floer homology to spaces associated to Heegaard diagrams. Although this theory is conjecturally isomorphic to Seiberg-Witten theory, it is more topological and combinatorial in flavor and thus easier to work with in certain contexts. The interaction between gauge theory, low-dimensional topology, and symplectic geometry has led to a number of striking new developments in these fields. The aim of this volume is to introduce graduate students and researchers in other fields to some of these exciting developments, with a special emphasis on the very fruitful interplay between disciplines. This volume is based on lecture courses and advanced seminars given at the 2004 Clay Mathematics Institute Summer School at the Alfred Renyi Institute of Mathematics in Budapest, Hungary. Several of the authors have added a considerable amount of additional material to that presented at the school, and the resulting volume provides a state-of-the-art introduction to current research, covering material from Heegaard Floer homology, contact geometry, smooth four-manifold topology, and symplectic four-manifolds.
Categories: Mathematics

Mathematics Journal of Toyama University

Mathematics Journal of Toyama University

There were also found many applications erential equations, topology, physics
and others. In this landscape udy of contact manifolds brought a lot of new
interesting results. n the subject of contact geometry there is also the class of
contact ...

Author:

Publisher:

ISBN: UCAL:B5271797

Category: Mathematics

Page:

View: 682

Categories: Mathematics

Reviews in Mathematics and Mathematical Physics

Reviews in Mathematics and Mathematical Physics

Fiz . 1989 . V . 78 . No . 1 . P . 136 - 139 ; English transl . in Theoret . Math . Phys .
1989 . V . 78 . No . 1 . P . 97 - 99 . ( 107 ] Mokhov O . I . Contact geometry and
calculus of variations . In : Geometry , Topology , and Applications . Moscow . MIP
.

Author:

Publisher:

ISBN: UOM:39015053961150

Category: Mathematical physics

Page:

View: 630

Categories: Mathematical physics

Singularities of Caustics and Wave Fronts

Singularities of Caustics and Wave Fronts

.'; '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.

Author: Vladimir Arnold

Publisher: Springer Science & Business Media

ISBN: 9789401133302

Category: Mathematics

Page: 259

View: 801

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.
Categories: Mathematics

Notices of the American Mathematical Society

Notices of the American Mathematical Society

... differential equations , semigroups of operators , control theory , mathematical
physics and related areas and applications Place : University ... geometric
foliations and laminations Place : Ecole Normale Supérieure de Lyon ( France )
For information contact : Frédéric ... Algebraic K - theory and its applications to
algebra , algebraic geometry , topology and number theory ; cyclic homology ,
homology of ...

Author: American Mathematical Society

Publisher:

ISBN: UCSD:31822020431672

Category: Mathematics

Page:

View: 964

Categories: Mathematics

Introduction to Smooth Manifolds

Introduction to Smooth Manifolds

In this final chapter we introduce a new kind of geometric structure on manifolds,
called a symplectic structure, which is ... Symplectic structures have surprisingly
varied applications in mathematics and physics, including partial differential
equations, differential topology, and classical mechanics, among many other
fields. ... Then at the end of the chapter, we show how symplectic and contact
geometry can be used to construct solutions to first-order partial differential
equations. J.M. Lee ...

Author: John Lee

Publisher: Springer Science & Business Media

ISBN: 9781441999825

Category: Mathematics

Page: 708

View: 129

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.
Categories: Mathematics