arXiv subjects

Dan Cristofaro-Gardiner

Publications and source records attributed to Dan Cristofaro-Gardiner.

23 records · Page 2Linked to original sources

arXiv2020

Special eccentricities of rational four-dimensional ellipsoids

A striking result of McDuff and Schlenk asserts that in determining when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional symplectic ball, the answer is governed by an "infinite staircase" determined by the odd-index Fibonacci numbers and the Golden Mean. Here we study embeddings of one four-dimensional symplectic ellipsoid into another, and we show that if the target is rational, then the infinite staircase phenomenon found by McDuff and Schlenk is quite rare. Specifically, in the rational case, there is an infinite staircase in precisely three cases -- when the target has "eccentricity" 1, 2, or 3/2; in all other cases the answer is given by the classical volume obstruction except on finitely many compact intervals on which it is linear. This verifies in the special case of ellipsoids a conjecture by Holm, Mandini, Pires, and the author.

math.SG↗

arXiv2019

Symplectic embeddings from concave toric domains into convex ones

Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. In "Symplectic embeddings into four-dimensional concave toric domains", the author, Choi, Frenkel, Hutchings and Ramos computed the ECH capacities of all "concave toric domains", and showed that these give sharp obstructions in several interesting cases. We show that these obstructions are sharp for all symplectic embeddings of concave toric domains into "convex" ones. In an appendix with Choi, we prove a new formula for the ECH capacities of convex toric domains, which shows that they are determined by the ECH capacities of a corresponding collection of balls.

math.SG↗

arXiv2018

Torsion contact forms in three dimensions have two or infinitely many Reeb orbits

We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it follows that under the above assumptions, there are infinitely many simple Reeb orbits if the three-manifold is not the three-sphere or a lens space. We also show that for non-torsion contact structures, every nondegenerate contact form has at least four simple Reeb orbits.

math.SG↗

arXiv2017

The ghost stairs stabilize to sharp symplectic embedding obstructions

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 ideas of independent interest, namely: (i) an improved version of the index inequality familiar from the theory of embedded contact homology (ECH), (ii) new applications of relative intersection theory in the context of neck stretching analysis, (iii) a new approach to estimating the ECH grading of multiply covered elliptic orbits in terms of areas and continued fractions, and (iv) a new technique for understanding the ECH of ellipsoids by constructing explicit bijections between certain sets of lattice points.

math.SG↗

arXiv2015

New examples of period collapse

"Period collapse" refers to any situation where the period of the Ehrhart function of a polytope is less than the denominator of that polytope. We study several interesting situations where this occurs, primarily involving triangles. For example: 1) we determine exactly when the Ehrhart function of a right triangle with legs on the axes and slant edge with irrational slope is a polynomial; 2) we find triangles with periods given by any even-index k-Fibonacci number, and larger denominators; 3) we construct several higher dimensional examples. Several related issues are also discussed, including connections with symplectic geometry.

math.CO↗