Embedding Problems in Symplectic Geometry

Let ω0 denotes the canonical symplectic 2-form on R2n. A smooth map φ defined on an open subset U of R2n is called symplectic if φ*ω0 = ω0. A symplectic map is called a symplectic embedding if it is injective. The study of symplectic embeddings is the main goal of this book. It is a relatively new, interesting and promising topic that has not existed for more than two decades. The book has nine chapters and five appendices but the main information is contained within chapter 1. Roughly speaking, all the other chapters contain proofs of the results announced in chapter 1 and further development of the theory. The main attention is devoted to various symplectic embedding constructions (lifting, wrapping and folding). These relatively elementary constructions are then used for proofs of various symplectic embedding results.
The prerequisites required for reading this book are very modest. One must be familiar with the notion of a differentiable manifold and with differential forms on manifolds. In a way, the whole theory is not at all easy and it requires a lot of techniques. Nevertheless the author‘s presentation is excellent and I do not think it could be written better. The book deals with a new and young area. Consequently, it contains many open problems and conjectures. The author mentions that it is addressed to mathematicians interested in geometry and dynamics and to physicists working in a field related to symplectic geometry. I think it will be interesting in particular for young mathematicians who will discover a new, wide and attractive area for their research.

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