Lectures on Symplectic Geometry (Lecture Notes in Mathematics, 1764) by Ana Cannas da Silva (PDF)

Description

These notes approximately transcribe a 15-week course on symplectic geometry I taught at UC Berkeley in the Fall of 1997. The course at Berkeley was greatly inspired in content and style by Victor Guillemin, whose masterly teaching of beautiful courses on topics related to s- plectic geometry at MIT, I was lucky enough to experience as a graduate student. I am very thankful to him! That course also borrowed from the 1997 Park City summer courses on symplectic geometry and topology, and from many talks and discussions of the symplectic geometry group at MIT. Among the regular participants in the MIT – formal symplectic seminar 93-96, I would like to acknowledge the contributions of Allen Knutson, Chris Woodward, David Metzler, Eckhard Meinrenken, Elisa Prato, Eugene Lerman, Jonathan Weitsman, Lisa Jeffrey, Reyer Sjamaar, Shaun Martin, Stephanie Singer, Sue Tolman and, last but not least, Yael Karshon. Thanks to everyone sitting in Math 242 in the Fall of 1997 for all the c- ments they made, and especially to those who wrote notes on the basis of which I was better able to reconstruct what went on: Alexandru Scorpan, Ben Davis, David Martinez,DonBarkauskas,EzraMiller,HenriqueBursztyn,John-PeterLund,Laura De Marco, Olga Radko, Peter P? rib´ ?k, Pieter Collins, Sarah Packman, Stephen Bigelow, Susan Harrington, Tolga Etgu ¨ and Yi Ma.

User’s Reviews

Editorial Reviews: Review “I find this to be both the best introduction to symplectic geometry as well as a model for how to introduce any field of study. … one feels the hand of a master in the text’s homework sets: concrete, illustrative, and enhancing the material presented. … For an upper-level undergraduate or beginning graduate student, Lectures on Symplectic Geometry remains, in my opinion, an ideal starting point into an exciting, active and growing area of mathematics.” (Andrew McInerney, MAA Reviews, June, 2018)From the reviews of the first printingOver 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.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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.

⭐I too agree with the previous reviewer that this is the best way for any graduate student in geometry/topology to learn symplectic geometry. Do not get me wrong, this is not an easy book to read, and the prerequisites aren’t low– you should be well acquainted with basic differential topology and differential geometry from a global viewpoint. But if you aren’t well acquainted with them to begin with, why are you picking up a text on symplectic geometry? You should learn those subjects first.This is not a comprehensive text– proofs of major theorems (MWM theorem comes to mind) are only sketched out, and the exposition of the moment map is very difficult on the initial read through (you need to take the definition and its properties by faith until you hit its formal development using Lie algebra cohomology in section 26), to name two problems with the text. However, working through the many exercise sets scattered through the book helps you develop confidence and geometric intuition with the objects dealt with, which is a major plus for this book.Graduate students who are working through this text should supplement this with a copy of a good differential geometry text with emphasis on differential forms (I recommend Morita’s “Geometry of Differential Forms” for the rudiments) and a lot of coffee. This isn’t an easy text, but then again, this isn’t an easy field.

⭐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.

⭐One of the most outstanding books on this subject.

⭐Sehr umfangreiche Einführung in die symplektische Geometrie, an vielen Stellen allerdings etwas knapp. Etwas ausfürhlicherere Erläuterungen täten oft gut. Auch der Aufbau von Definitionen ist stellenweise etwas gewöhnungsbedürftig.

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