Constructive Mathematics by Richman F. (ed.)

By Richman F. (ed.)

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Notice that because a, b, a are areas the volume obstruction to the existence of an embedding is that ab ≤ (a )2 . This question was nicely formulated in a paper by Cieliebak, Hofer, Latschev and Schlenk [3] called Quantitative Symplectic Geometry in terms of the following function: define c(a) for a ≥ 1 byh c(a) := inf{c : E(a, b) embeds symplectically in E(c , c )}. g We say that the set U embeds symplectically in V if there is a symplectomorphism φ such that φ(U ) ⊂ V . h Note that E(a, b) may not embed in E c(a), c(a) itself – one usually needs a little extra room so that the boundary of E(a, b) does not fold up on itself.

In k+1 (x)y i1 . . y in O(x; y) = n=0 ∂ |α1 | ∂ |αk+1 | , ⊗ . . ⊗ ∂y α1 ∂y αk+1 (31) November 4, 2009 30 13:57 WSPC - Proceedings Trim Size: 9in x 6in ewmproc S. in k+1 (x) are coefficients of tensors symmetric in the covariant indices i1 , . . , in . and symmetric in each block of αi contravariant indices. The spaces Ω(M, Tpoly ) and Ω(M, Dpoly ) have a formal fiberwise DGLA structure: the degree of an element in Ω(M, Tpoly ) ( resp. Ω(M, Dpoly ) ) is defined by the sum of the degree of the exterior form and the degree of the polyvector field (resp.

When [3] was written, this function was largely a mystery except that one knew that c(a) = a for a ≤ 2 (rigidity). Now methods have been developed to understand it, and it should be fully known soon for all a: see McDuff and Schlenk [12] and also [11]. As a first step, work of Opshtein [14] can be used to evaluate c(a) in the range 1 ≤ a ≤ 4. Surprisingly, it turns out that c(a) = a, if 1 ≤ a ≤ 2, c(a) = 2 if 2 ≤ a ≤ 4. (2) In other words the graph is constant in the range a ∈ [2, 4]. To prove this one only needs to show that c(4) = 2.

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