Ellipsoid embeddings and symplectic packing stability

Buse, Olguta; Hind, Richard

doi:10.1112/s0010437x12000826

Cited by 16 publications

(23 citation statements)

References 21 publications

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“…Symplectic ellipsoids are the main heroes of this story. This is clear for this text in view of our choice (1.1), but also for other recent advances on symplectic embeddings, such as packing stability in higher dimensions [17,18] and the connections between symplectic embedding problems and lattice point counting [27,28], ellipsoids play a key role.…”

Section: Symplectic Ellipsoidsmentioning

confidence: 84%

“…This text only touches upon a few of these connections and a few of the new results on symplectic embeddings. Among the unforgivable omissions are the breakthrough in the problem of ball packing stability in higher dimensions by Buse and Hind [17,18] and Hutchings's ECH capacities [62], which form a whole sequence of symplectic embedding invariants of 4-dimensional domains, that provide a complete set of obstructions for many embedding problems. Excellent surveys on ECH capacities are [63,64], and a quite comprehensive survey on the new results produced by the third revolution is [98].…”

Section: E(1 A)mentioning

confidence: 99%

“…. For 2n 6 the numbers p 2n k,S are not known in general, but by packing stability [18] there are constants k 0 (2n) such that p 2n k,S = 1 for all k k 0 (2n). This is very much larger than p 2n k,E for large k by (5.2).…”

Section: Three Ways To Pack a Boxmentioning

confidence: 99%

“…It is known that p k (C 6 ) = k 6 for k 6[66] and that p k (C 6 ) = 1 for k of the form 6 3[81] or for k k 0 with a nonexplicit constant k 0[18].…”

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confidence: 99%

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Symplectic embedding problems, old and new

Schlenk

2017

Bull. Amer. Math. Soc.

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Section: Symplectic Ellipsoidsmentioning

confidence: 84%

Section: E(1 A)mentioning

confidence: 99%

Section: Three Ways To Pack a Boxmentioning

confidence: 99%

“…It is known that p k (C 6 ) = k 6 for k 6[66] and that p k (C 6 ) = 1 for k of the form 6 3[81] or for k k 0 with a nonexplicit constant k 0[18].…”

mentioning

confidence: 99%

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Symplectic embedding problems, old and new

Schlenk

2017

Bull. Amer. Math. Soc.

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“…In general, determining which special four‐dimensional features have higher dimensional analogues is a central question in symplectic geometry. Other constructions embedding higher dimensional ellipsoids, which are sometimes optimal, can be found in .…”

Section: Introductionmentioning

confidence: 99%

The ghost stairs stabilize to sharp symplectic embedding obstructions

Cristofaro-Gardiner

Hind

McDuff

2018

Journal of Topology

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In determining when a four-dimensional ellipsoid can be symplectically embedded into a ball, McDuff and Schlenk found an infinite sequence of 'ghost' obstructions that generate an infinite 'ghost staircase' determined by the even index Fibonacci numbers. The ghost obstructions are not visible for the four-dimensional embedding problem because strictly stronger obstructions also exist. We show that in contrast, the embedding constraints associated to the ghost obstructions are sharp for the stabilized problem; moreover, the corresponding optimal embeddings are given by symplectic folding.The proof introduces several techniques of independent interest, namely: (i) new applications of ideas from embedded contact homology (ECH) in the context of nodal curves, (ii) an improved version of the index inequality familiar from the theory of embedded contact homology, (iii) new applications of relative intersection theory in the context of neck stretching analysis, (iv) a new approach to estimating the ECH grading of multiply covered elliptic orbits in terms of areas and continued fractions, and (v) a new technique for understanding the ECH of ellipsoids by constructing explicit bijections between certain sets of lattice points.Contents Actually the construction in [8] only applies to compact subsets of the stabilized ellipsoid. To embed the whole product we are appealing to [23].q with gcd(p, q) = 1 and ε > 0 irrational and very small, and fix μ * > 0. Suppose that for all sufficiently small ε > 0 and generic admissible J there is a genus 0 curve C in X μ ,x with degree d, Fredholm index 0, and one negative end on {(β 1 , p)}, where gcd(d, p) = 1. Then, 3d = p + q, and for all k 0, we haveThe proof is given in § 3.6: see Proposition 3.6.1. The main point is that if C has genus 0, Fredholm index zero, and one negative end, then its Fredholm index remains zero under † In fact if the domain is an ellipsoid the two embedding problems are equivalent: see [19]. Here CP 2 (μ) denotes CP 2 with symplectic form ω scaled so that ω takes the value μ on the line L.‡ Here we assume that J is admissible, that is, adapted to the negative end of X: see § 2 for more details. † In fact, the ECH cobordism map detects buildings with ECH index 0 that may (and often do) consist of curves with both positive and negative ECH index. Further, it may not always be the case that curves with ECH index 2 and Fredholm index 0 must have double points, but as mentioned in Remark 2.2.3(ii) this is known in some situations.

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Irreversibility from staircases in symplectic embeddings

Creaco

Kalogeropoulos

2019

Physica A: Statistical Mechanics and its Applications

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We present an argument whose goal is to trace the origin of the macroscopically irreversible behavior of Hamitonian systems of many degrees of freedom. We use recent flexibility and rigidity results of symplectic embeddings, quantified via the (stabilized) Fibonacci and Pell staircases, to encode the underlying breadth of the possible initial conditions, which alongside the multitude of degrees of freeedom of the underlying system give rise to time-irreversibility.

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Ellipsoid embeddings and symplectic packing stability

Cited by 16 publications

References 21 publications

Symplectic embedding problems, old and new

Symplectic embedding problems, old and new

The ghost stairs stabilize to sharp symplectic embedding obstructions

Irreversibility from staircases in symplectic embeddings

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