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.
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From the reviews of the first printing Over the years, there have been several books written to serve as an introduction to symplectic geometry and topology, […] The text under review here fits well within this tradition, providing a useful and effective synopsis of the basics of symplectic geometry and possibly serving as the springboard for a prospective researcher. The material covered here amounts to the "usual suspects" of symplectic geometry and topology. From an introductory chapter of symplectic forms and symplectic algebra, the book moves on to many of the subjects that serve as the basis for current research:symplectomorphisms, Lagrangian submanifolds, the Moser theorems, Darboux-Moser-Weinstein theory, almost complex structures, Kãhler structures, Hamiltonian mechanics, symplectic reduction, etc. The text is written in a clear, easy-to-follow style, that is most appropriate in mathematical sophistication for second-year graduate students; […]. This text had its origins in a 15-week course that the author taught at UC Berkeley. There are some nice passages where the author simply lists some known results and some well-known conjectures, much as one would expect to see in a good lecture on the same subject. Particularly eloquent is the author’s discussion of the compact examples and counterexamples of symplectic, almost complex, complex and Kähler manifolds. Throughout the text, she uses specific, well-chosen examples to illustrate the results. In the initial chapter, she provides a detailed section on the classical example of the syrnplectic structure of the cotangent bundle of a manifold. Showing a good sense of pedagogy, the author often leaves these examples as well-planned homework assignments at the end of some of the sections. […] In all of these cases, the author gives the reader a chance to illustrate and understand the interesting results of each section, rather than relegating the tedious but needed results to the reader. Mathematical Reviews 2002i
From the Back Cover
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.
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4 of 4 people found the following review helpful:
4.0 out of 5 stars
the most accessible text on the subject,
This review is from: Lectures on Symplectic Geometry (Lecture Notes in Mathematics) (Paperback)
I disagree with the previous reviewer. This is the most accessible text I know on symplectic geometry, and the one I would recommend to a beginning graduate student trying to learn the topic on their own. It is certainly easier (and lighter) than McDuff and Salamon's book (Introduction to Symplectic Topology), which is much more comprehensive but requires more effort.
6 of 10 people found the following review helpful:
3.0 out of 5 stars
Difficult as to be expected...,
By Cybertronian (Finland) - See all my reviews
This review is from: Lectures on Symplectic Geometry (Lecture Notes in Mathematics) (Paperback)
On the one hand, it's a good and fairly complete book on modern symplectic geometry and Hamiltonian systems, but on the other hand it is extremely dense without any illuminating examples, which makes reading it very difficult. Unfortunately, the concise lecture notes style is typical in mathematics today, so there aren't many alternatives if you want a book that deals with most of the theory.
In order to read and understand the book you probably have to be an advanced graduate student in mathematics with a clear affinity towards minimalistic, sentence-free lemma-proof-theorem-proof texts. As a textbook I cannot recommend these Lectures on Symplectic Geometry, as they are too dense, too proof-oriented and too example-devoid. As a quick reference for young researchers in mathematics who already master the rudiments of symplectic topology and geometry it might be an inexpensive possibility. For all others (the actually intended audience, i.e. the interested students who want to learn about symplectic geometry) it is a non-risk-free investment: you have to take the time and go very slowly through the material and either you are rewarded for your great efforts eventually or you are stuck at trivialities and become discouraged unduly.
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Inside This Book
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First Sentence:
A symplectic form is a 2-form satisfying an algebraic condition - nondegeneracy - and an analytical condition - closedness. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
reduced symplectic form, toric manifolds, symplectic vector fields, moment polytope, tubular neighborhood theorem, moment map, symplectic vector space, hamiltonian way, hamiltonian action, symplectic action, compact symplectic manifold, symplectic basis, minimizing geodesic, positive inner product, contact manifold, equivariant cohomology, lagrangian submanifolds, symplectic manifolds, lagrangian subspace, symplectic forms, symplectic geometry, circle bundles, slice theorem, complex submanifold, hamiltonian vector field New!
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A symplectic form is a 2-form satisfying an algebraic condition - nondegeneracy - and an analytical condition - closedness. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
reduced symplectic form, toric manifolds, symplectic vector fields, moment polytope, tubular neighborhood theorem, moment map, symplectic vector space, hamiltonian way, hamiltonian action, symplectic action, compact symplectic manifold, symplectic basis, minimizing geodesic, positive inner product, contact manifold, equivariant cohomology, lagrangian submanifolds, symplectic manifolds, lagrangian subspace, symplectic forms, symplectic geometry, circle bundles, slice theorem, complex submanifold, hamiltonian vector field New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
Citations (learn more)
This book cites 75
books:
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56
books
cite this book:
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- Forward-Backward Stochastic Differential Equations and their Applications (Lecture Notes in Mathematics) by Jin Ma in Back Matter (1), and Back Matter (2)
- Differential Topology by Victor Guillemin in Back Matter
- Partial Differential Relations (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics) by Mikhael Gromov in Back Matter
- Grothendieck Duality and Base Change (Lecture Notes in Mathematics) by Brian Conrad in Back Matter
- Differential Forms (Dover Books on Mathematics) by Henri Cartan in Back Matter
See all 75 books this book cites
- Geometric Mechanics by Waldyr Muniz Oliva in Back Matter (1), and Back Matter (2)
- hp-Finite Element Methods for Singular Perturbations (Lecture Notes in Mathematics) by Jens M. Melenk in Back Matter
- Big Queues (Lecture Notes in Mathematics) by Ayalvadi J. Ganesh in Back Matter
- Geometry of the Plane Cremona Maps (Lecture Notes in Mathematics) by Maria Alberich-Carraminana in Back Matter
- Geometric Aspects of Functional Analysis: Israel Seminar 2001-2002 (Lecture Notes in Mathematics) by Vitali D. Milman in Back Matter
See all 56 books citing this book
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