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**Quantitative symplectic geometry.**
*(English)*
Zbl 1143.53341

Hasselblatt, Boris (ed.), Dynamics, ergodic theory and geometry. Dedicated to Anatole Katok. Based on the workshop on recent progress in dynamics, Berkeley, CA, USA, from late September to early October, 2004. Cambridge: Cambridge University Press (ISBN 978-0-521-87541-7/hbk). Mathematical Sciences Research Institute Publications 54, 1-44 (2007).

From the authors’ introduction: Symplectic capacities were introduced in 1990 by I. Ekeland and H. Hofer [Math. Z. 200, 355–378 (1989; Zbl 0641.53035); ibid. 203, 553–567 (1990; Zbl 0729.53039)] (although the first capacity was in fact constructed by M. Gromov [Invent. Math. 82, 307–347 (1985; Zbl 0592.53025)]). Since then, lots of new capacities have been defined.

Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. This is illustrated in Section 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In Section 3, we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities. In Section 4, we describe several new relations between certain symplectic capacities on ellipsoids and polydiscs. Throughout the discussion, we mention many open problems.

For the entire collection see [Zbl 1128.37001].

Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. This is illustrated in Section 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In Section 3, we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities. In Section 4, we describe several new relations between certain symplectic capacities on ellipsoids and polydiscs. Throughout the discussion, we mention many open problems.

For the entire collection see [Zbl 1128.37001].

### MSC:

53D05 | Symplectic manifolds (general theory) |