Introduction To Symplectic Geometry

Author : Rolf Berndt
Publisher : American Mathematical Society
ISBN 13 : 1470476886
Total Pages : 213 pages
Book Rating : 4.85/5 ( download)

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Download or read book An Introduction to Symplectic Geometry written by Rolf Berndt and published by American Mathematical Society. This book was released on 2024-04-15 with total page 213 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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, Kähler 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.

Author : Ana Cannas da Silva
Publisher : Springer
ISBN 13 : 354045330X
Total Pages : 220 pages
Book Rating : 4.07/5 ( download)

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Download or read book Lectures on Symplectic Geometry written by Ana Cannas da Silva and published by Springer. This book was released on 2004-10-27 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.

Author : Jean-Louis Koszul
Publisher : Springer
ISBN 13 : 9811339872
Total Pages : 121 pages
Book Rating : 4.75/5 ( download)

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Download or read book Introduction to Symplectic Geometry written by Jean-Louis Koszul and published by Springer. This book was released on 2019-04-15 with total page 121 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introductory book offers a unique and unified overview of symplectic geometry, highlighting the differential properties of symplectic manifolds. It consists of six chapters: Some Algebra Basics, Symplectic Manifolds, Cotangent Bundles, Symplectic G-spaces, Poisson Manifolds, and A Graded Case, concluding with a discussion of the differential properties of graded symplectic manifolds of dimensions (0,n). It is a useful reference resource for students and researchers interested in geometry, group theory, analysis and differential equations.This book is also inspiring in the emerging field of Geometric Science of Information, in particular the chapter on Symplectic G-spaces, where Jean-Louis Koszul develops Jean-Marie Souriau's tools related to the non-equivariant case of co-adjoint action on Souriau’s moment map through Souriau’s Cocycle, opening the door to Lie Group Machine Learning with Souriau-Fisher metric.

Author : Dusa McDuff
Publisher : Oxford University Press
ISBN 13 : 0198794894
Total Pages : 637 pages
Book Rating : 4.99/5 ( download)

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Download or read book Introduction to Symplectic Topology written by Dusa McDuff and published by Oxford University Press. This book was released on 2017 with total page 637 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. This new third edition of a classic book in the feild includes updates and new material to bring the material right up-to-date.

Author : Maurice A. de Gosson
Publisher : Springer Science & Business Media
ISBN 13 : 3764375752
Total Pages : 375 pages
Book Rating : 4.51/5 ( download)

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Download or read book Symplectic Geometry and Quantum Mechanics written by Maurice A. de Gosson and published by Springer Science & Business Media. This book was released on 2006-08-06 with total page 375 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.

Author : Katharina Habermann
Publisher : Springer
ISBN 13 : 3540334211
Total Pages : 131 pages
Book Rating : 4.17/5 ( download)

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Download or read book Introduction to Symplectic Dirac Operators written by Katharina Habermann and published by Springer. This book was released on 2006-10-28 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the first one that gives a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflects the current state of the subject. At the same time, it is intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology, which have become important fields and very active areas of mathematical research.

Author : Yakov Eliashberg
Publisher : American Mathematical Soc.
ISBN 13 : 9780821886892
Total Pages : 452 pages
Book Rating : 4.94/5 ( download)

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Download or read book Symplectic Geometry and Topology written by Yakov Eliashberg and published by American Mathematical Soc.. This book was released on 2004 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introductionto Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristicsand Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden. Information for our distributors: Titles in this series are copublished with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Author : P. Libermann
Publisher : Springer Science & Business Media
ISBN 13 : 9400938071
Total Pages : 541 pages
Book Rating : 4.76/5 ( download)

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Download or read book Symplectic Geometry and Analytical Mechanics written by P. Libermann and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 541 pages. Available in PDF, EPUB and Kindle. Book excerpt: Approach your problems from the right end It isn't that they can't see the solution. and begin with the answers. Then one day, It is that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' Brown 'The point of a Pin'. in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thouglit to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sci ences has changed drastically in recent years: measure theory is used (non-trivially) in re gional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homo topy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

Author : Franco Cardin
Publisher : Springer
ISBN 13 : 3319110268
Total Pages : 237 pages
Book Rating : 4.64/5 ( download)

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Download or read book Elementary Symplectic Topology and Mechanics written by Franco Cardin and published by Springer. This book was released on 2014-12-01 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this rich framework: analytical mechanics, the calculus of variations, conjugate points & Morse index, and other physical topics. A central feature is the systematic utilization of Lagrangian submanifolds and their Maslov-Hörmander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of Hamilton-Jacobi equations, Hamiltonian vector fields and canonical transformations are described by suitable Lagrangian submanifolds belonging to distinct well-defined symplectic structures. This unified point of view has been particularly fruitful in symplectic topology, which is the modern Hamiltonian environment for the calculus of variations, yielding sharp sufficient existence conditions. This line of investigation was initiated by Claude Viterbo in 1992; here, some primary consequences of this theory are exposed in Chapter 8: aspects of Poincaré's last geometric theorem and the Arnol'd conjecture are introduced. In Chapter 7 elements of the global asymptotic treatment of the highly oscillating integrals for the Schrödinger equation are discussed: as is well known, this eventually leads to the theory of Fourier Integral Operators. This short handbook is directed toward graduate students in Mathematics and Physics and to all those who desire a quick introduction to these beautiful subjects.

Author : Michèle Audin
Publisher : Birkhäuser
ISBN 13 : 3034880715
Total Pages : 225 pages
Book Rating : 4.18/5 ( download)

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Download or read book Symplectic Geometry of Integrable Hamiltonian Systems written by Michèle Audin and published by Birkhäuser. This book was released on 2012-12-06 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Among all the Hamiltonian systems, the integrable ones have special geometric properties; in particular, their solutions are very regular and quasi-periodic. This book serves as an introduction to symplectic and contact geometry for graduate students, exploring the underlying geometry of integrable Hamiltonian systems. Includes exercises designed to complement the expositiont, and up-to-date references.