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**Explicit symplectic packings. Symplectic tunnelling and new maximal constructions.**
*(English)*
Zbl 1169.52009

Berichte aus der Mathematik. Aachen: Shaker Verlag; Köln: Univ. Köln (Diss. 2008) (ISBN 978-3-8322-7954-7/hbk). xviii, 309 p. (2009).

This book is author’s Ph. D. dissertation with added introductory chapter which discusses the Euclidean ball packing problem and some related topics.

Symplectic geometry is distinguished from volume geometry by its embedding obstructions, which lead to symplectic invariants like the ball packing widths. Although these can be implicitly computed for some important manifolds, explicit constructions are required to develop a geometric understanding of the arising obstructions.

The main part of this book discusses methods that admit the construction of explicit packings. The previously established approaches of deformation and wrapping are shortly reviewed and then thoroughly analyzed to determine their potential and limitations. With the introduction of symplectic tunnelling, a new effective method is developed.

As an application of these methods, explicit solutions are constructed for specific packing problems. Besides many previously unachievable explicit maximal packings of ruled symplectic manifolds, a major result is the construction of the long-sought maximal 7- and 8-packings of the four dimensional symplectic ball, which the method of tunnelling affords.

In the part I are summarized the necessary background on symplectic geometry and the ball packing widths. In the part II are discussed methods that admit the construction of explicit packings, while in the part III are applied these methods to construct solutions for specific problems.

Symplectic geometry is distinguished from volume geometry by its embedding obstructions, which lead to symplectic invariants like the ball packing widths. Although these can be implicitly computed for some important manifolds, explicit constructions are required to develop a geometric understanding of the arising obstructions.

The main part of this book discusses methods that admit the construction of explicit packings. The previously established approaches of deformation and wrapping are shortly reviewed and then thoroughly analyzed to determine their potential and limitations. With the introduction of symplectic tunnelling, a new effective method is developed.

As an application of these methods, explicit solutions are constructed for specific packing problems. Besides many previously unachievable explicit maximal packings of ruled symplectic manifolds, a major result is the construction of the long-sought maximal 7- and 8-packings of the four dimensional symplectic ball, which the method of tunnelling affords.

In the part I are summarized the necessary background on symplectic geometry and the ball packing widths. In the part II are discussed methods that admit the construction of explicit packings, while in the part III are applied these methods to construct solutions for specific problems.

Reviewer: Milica Stojanović (Beograd)

### MSC:

52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |

52C15 | Packing and covering in \(2\) dimensions (aspects of discrete geometry) |

53D35 | Global theory of symplectic and contact manifolds |

52-02 | Research exposition (monographs, survey articles) pertaining to convex and discrete geometry |