An Introduction to Symplectic Geometry (Graduate Studies in Mathematics) (Graduate Studies in Mathematics) - Hardcover
Book 19 of 190: Graduate Studies in Mathematics
Synopsis
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.
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Language Notes
Text: English (translation)
Original Language: German
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An Introduction to Symplectic Geometry
Published by
American Mathematical Society, Providence, RI, 2001
ISBN 10: 0821820567
ISBN 13: 9780821820568
Used
Hardcover
First Edition
Seller: Black Cat Hill Books, Oregon City, OR, U.S.A.
Seller rating 5 out of 5 stars
Hardcover. Condition: Near Fine. First Edition; Second Printing. Very Near Fine: shows only a superficial blow to the upper fore-edge causing a very faint wrinkle to about 35 pages late in the text; else flawless; the binding is square and secure; the text is clean. Free of any underlining, hi-lighting or marginalia or marks in the text. Free of ownership names, dates, addresses, notations, inscriptions, stamps, or labels. A handsome, nearly-new copy, structurally sound and tightly bound, showing a minor, unobtrusive imperfection. Bright and clean. Corners sharp. Very close to 'As New'. NOT a Remainder, Book-Club, or Ex-Library. Small 4to (10.3 x 7.25 x 0.7 inches) . Language: English. Weight: 19.3 ounces. First Edition (2001) , Second Printing (2009). Graduate Studies in Mathematics Series, No. 26. Hardcover: Laminate Boards. 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. ; Graduate Studies in Mathematics; Vol. 26; Small 4to 9" - 11" tall; xvi, 195 pages. Seller Inventory # 59216